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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at http : //books . google . com/| Philosophical transactions of the Royal Society of London Royal Society (Great Britain), JSTOR (Orgariizatiori) Jigitized by Google Digitized by VjOOQ IC Digitized by VjOOQ IC R PHILOSOPHICAL TRANSACTIONS. / {^ ■' ' :i OF THE ROYAL SOCIETY OF LONDON. Seiiies a. CONTAINING PAPEES OF A MATHEMATICAL OK PHYSICAL CHAKACTEE. VOL. 195. LONDON: IMJfNTED I5Y IIARKISON AM) SONS, ST. MARTINS LANB, W.O., |)rinttTS in #rbinBni lo $ti Pajtsig. January, 1901, Digitized by VjOOQ IC Digitized by Google C m ] CONTENTS. (A) VOL. 195. List of Illustrations page ▼ Advertisement vii List of Institutions entitled to receive the Philosophical Transactions or Proceedings of the Royal Society ' ix Adjudication of Medals xvii X T. Mathematical Coritribntions to the Theory of Evolution, — VIl. On the Con'elatio7i of Characters not Quantitatively Measurable. By Karl Peaiison, F.Ii.S. page 1 11. Electrical Conductivity in Gases Traversed by Cathode Rays. By J. C. McLennan, Demonstrator in Physics^ University of Toronto. Communicated by Professor J. J. Thomson, F.R.S. 49 ^ 111. Mathematical Contnbutions to the Theory of Evolution. — VIII. On the Inheri- tance of Character's not capable of Exact Qvxintitative Meamrement. — Part I. Introductory. Part II. On the Inheritance of Coat-colour in Horses. Part III. On the Inheritance of Eye-colour in Man. By Kabl Peabson, F.R.S. , ivith the assistance of Alice Lee, D.Sc.y University College, London 79 IV. On Simultaneous Partial Differential Equations. By A, C. Dixon, Sc.D. Communicoied by J. W. L. Glaibheu, Sc.D 151 a 2 Digitized by Google [ iv ] V. The Velocity of the Ions produced in Gases hy Ronlgen Rays. By John Zeleny, B.Sc.y B.A., Assistant Professor of Physics, University of Minnesota, Com- municated hy Professor J. J. Thomson, F.R.S. 193 VI. Underground Temperature at Oxford in the Yexxr 1899, as determined hy Five Platinum-resistance Thermometers. By Arthur A. Rambaut, M.A., D.Sc, Radcliffe Ohserver. Communicated hy E. H. Griffiths, F.R.S. . . . 235 VII. The Diffusion of Ions produced in Air hy the Action of a Radio-active SuhstancCy Ultra-violet Light and Point Discharges. By John S. Townsend, M.A., Clerk-Maxwell Student, Cavendish Laboratory^ Fellow of Trinity College^ Cambridge. Communicated hy Professor J. J. Thomson, F.R,S. 259 VIII. T/ie Crystalline Structure of Metals. (Second Paper.) By J. A. EwiNG, F.R.S. , Professor of Mechanism and Applied Mechanics in the University of Cambridge, and Walter Rosenhain, B.A., St. John's College, Cambridge^ \^5\ Exhibition Research Scholar, University of Melbourne .... 279 IX. Lilies of Induction in a Magnetic Field. By Professor H. S. Hele-Shaw, LL.D., F.R.S., and Alfred Hay, B.Sc 303 X. On tlie Application of Fourier's Double Integrals to Optical Problems. By Charles GtODFREY, B.A., Scholar of Trinity, Isaac Neioton Student in the University of Cambridge. Communicated by Professor J. J. Thomson, F.R.S. 329 XL An Experimental Investigation into the Flow of Marble. By Frank Dawson Adams, M.Sc, Ph.D., F.G.S., Logan Professor oj Geology in McGill University, and John Thomas Nicolson, D.Sc, M.Inst., C.E., Head of the Engineering Departmenty Manchester Municipal Technical School {formerly Professor of Mechanical Engi7ieeri7ig in McGill University) .... 363 Index to Volume 403 Erratum 405 Digitized by Google LIST OF ILLUSTRATIONS. Plates* 1 and 2. — Dr. A. A. Rambaut on the Underground Temperature at Oxford in the year 1899, as determined by Five Platinum-resistance Thermometers. Plates 3 to 13. — Professor J. A. Ewing and Mr. W. Rosenhain on the Crystalline Structure of Metals. Plates 14 to 21. — Professor H. S. Hele-Shaw and Mr. Alfred Hay on Lines of Induction in a Magnetic Field. Plates 22 to 25. — Professor F. D. Adams and Dr. J. T. Nioolson on an Experimental Investigation into the Flow of Marble. Digitized by VnOOQ iC Digitized by VjOOQ IC [ vii ] ADVERTISEMENT, The Committee appointed by the Royal Society to direct the publication of the Philosophical Transactions take this opportunity to acquaint the public that it fuDy appears^ as well from the Council-books and Journals of the Society as from repeated declarations which have been made in several former Transactions^ that the printing of them was always, from time to time, the single act of the respective Secretaries till the Forty-seventh Volume ; the Society, as a Body, never interesting themselves any further in their publication than by occasionally recommending the revival of them to some of their Secretaries, when, from the particular circumstances of their affairs, the Transactions had happened for any length of time to be intermitted. And this seems principally to have been done with a view to satisfy the public that their usual meetings were then continued, for the improvement of knowledge and benefit of mankind : the great ends of their first institution by the Royal Charters, and which they have ever since steadily pursued. But the Society being of late years greatly enlarged, and their communications more numerous, it was thought advisable that a Committee ot their members should be appointed to reconsider the papers read before them, and select out of them such as they should judge most proper for publication in the future Transactions ; which was accordingly done upon the 26th of March, 1752. And the grounds of their choice are, and will continue to be, the importance and singularity of the subjects, or the advantageous manner of treating them ; without pretending to answer for the certainty of the facts, or propriety of the reasonings contained in the several papers so published, which must still rest on the credit or judgment of their respective authors. It is likewise necessary on this occasion to remark, that it is an established rule of the Society, to which they will always adhere, never to give their opinion, as a Body, Digitized by VnOOQ iC [ ^^iii ] upon any subject, either of Nature or Art, that comes before them. And therefore the thanks, which are frequently proposed from the Chair, to be given to the authors of such papers as are read at their accustomed meetings, or to the persons through whose hands they received them, are to be considered in no other light than as a matter of civility, in return for the respect shown to the Society by those communications. The like also is to be said with regard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public newspapers, that they have met with the highest applause and approbation. And therefore it is hoped that no regard will hereafter be paid to such reports and public notices ; which in some instances have been too lightly credited, to the dishonour of the Society. Digitized by VnOOQ iC [ ix ] 1900. List of Institutions entitled to receive the Philosophical Transacteons or Proceedings of the Royal Society. Institutions marked a are entitled to receive Philosophical Transactions, Series A, and Proceedings. „ „ B „ „ „ „ Series B, and Proceedings. „ AB „ „ „ „ Series A and li, and Procee<ling«. „ „ /> „ „ Proceedings only. America (Central). Mexico. p. Sociedad Cientifica " Antonio Alzate." Ainerica(North). (See United States and Canada.) America (South). Buenos Ayres. A6. Mnseo Naclonal. Caracas. B. University Library Cordova. AB. Academia Nacional de Ciencias. Demerara. p, Hoyal Agricultural and Commercial Society, British Guiana. LiQ. Plata. B. Museo de La Plata. Rio de Janeiro. p. Observatorio. Australia. Adelaide. p. Royal Society of South Australia. Brisbane. ^. Royal Society of Queensland. Melbourne. p. Observatory. p. Royal Society of Victoria. AB. University Library. Sydney. Australian Museum. Geological Survey. Linnean Society of New South Wales. Royal Society of New South Wales. University Library. P- P- P- AB. AB* Austria. Agram. p, Jugoslavenska Akademija Znanosti i Um< jetnosti. p. Societas Historico-Naturalis Croatica. I VOL. CXCV. — A. b Anstria (continued). Briinn. AB. Naturforachender Verein. Gratz. AB. Naturwissenschaftlicher Verein fur Steier- mark. Innsbinick. AB. Das Ferdinandeum. p, Naturwissenschaftlich - Medicinischer Verein. Prague. AB. Konigliche Bohmisclie Gesellschaft dcr Wissenschaften. Trieste. B. Museo di Storia Naturale. p. Societa Adriatica di Scienze Naturali. Vienna. Antbropologische Gesellschaft. Kaiserliche Akademie der Wissenschaften. K.K. Geographische Gesellschaft. K.K. Geologische Reichsanstalt. K.K. Naturhistorisches Hof- Museum. K.K. Zoologisch-Botanische Gesellschaft. Oesterreichische Gesellschaft fiir Meteoro- logie. Von Kuffner'sche Stemwarte. Belgium. Brussels. B. Academic Roy ale de Medecine. Academic Royale des Sciences. Mus^ Royal d'Histoire Naturelle de Belgique. Observatoire Royal. Soci6t6 Beige de Geologic, de Paleonto- logie, et d'Hydrologie. Societe Malacologique de Belgique. P- aB. P' AB. B. B. P' A. AB. P- P' P' Ghent. AB. Univei-site. Digitized by Google \.. [ X J Belgium (continued). Li^ge. AB. Soci^te des Sciences. p. Soci^te Geologique de Belgique, Lonvain. B. Laboratoire de Microscopic et de Biologie Cellulaire, AB. Universite. Canada. Predericton, N.B. p, Universitj of New Brunswick. Halifax, N.S. p. Nova Scotian Institute of Science. Hamilton. p, Hamilton Association. Montreal. AB. McGill University. p. Natural History Society. Ottawa. AB. Geological Survey of Canada. AB. Royal Society of Canada. St. John, N.B. p. Natural History Society. Toronto. p. Toronto Astronomical Society. p, Canadian Institute. AB. University. Windsor, N.S. p. King's College Libi*ary. Cape of Oood Hope, A. Observatory. AB. South African Library. Ceylon. Colombo. B. Museum. China. Shanghai. p. China Branch of tlie Royal Asiatic Society. Denmark. Copenhagen. AB. Kongelige Danske Videnskabernes Selskab. Egypt. Alexandria. AB. Biblioth^que Municipale. England and Wales. Aberysiwith. AB. Univei-sity College. Bangor. AB. University College of Noith Wales. Birmingham. AB. Free Central Library. AB. Mason College. p. Philcsophical Society. England and Wales (continued). Bolton. p. Public Library. Bristol. p. Merchant Venturers' School. AB. Univei*8ity College. Cambridge. AB. Philosophical Society. p. Union Society. Cardiff. AB. University College. Cooper's Hill. AB. Royal Indian P]ngineering College. Dudley. p, Dudley and Midland Geological aud Scientific Society. Essex. p. Essex Field Club. Falmouth. p. Royal Cornwall Polytechnic Society. Greenwich. A. Royal Observatory. Kew. B. Royal Gardens. Leeds. p. Philosophical Society, AB. Yorkshire College. Liverpool. AB. Free Public Library. p, Literary and Philosophical Society. A. Observatory. AB. University College. London. AB. Admiralty. p. Anthropological Institute. AB. Board of Trade (Electrical Standards Laboratory). AB. British Museum (Nat. Hist.). AB. Chemical Society. A. City and Guilds of London Institute. p. '* Electrician," Editor of the. B. Entomological Society. AB. Geological Society. AB. Geological Survey of Gi'eat Britain. p. Geologists' Association. AB. Guildhall Library. A. Institution of Civil Engineers. p. Institution of Electrical Engineers. A. Institution of Mechanical Engineers. A. Institution of Naval Architects. p. Iron and Steel Institute. AB. King's College. B. Linnean Society. AB. London Institution. Digitized by Google [ xi ] , f 1 ^' England and Wales (continued). London (continued). London Library. Mathematical Society. Meteorological Office. Odontological Society. Pharmaceutical Society. Physical Society. Quckett Microscopical Club. Royal Agricultural Society. Royal Astronomical Society. Royal College of Physicians. Royal College of Surgeons. Royal Eiigiueera (for Libraries abroad, six copies). Royal Engineers. Head Quarters Library. Royal Geographical Society. Royal Horticultural Society. Royal Institute of British Architects. Royal Listitution of Great Britain. Royal Medical and Chirurgical Society. Royal Meteorological Society. Royal Microscopical Society. Royal Statistical Society. Royal United Service Institution. Society of Arts. Society of Biblical Archaeology. Society of Chemical Industry (London Section). Standard Weights and Measures Depaii- ment. The Queen's Library. The War Office. University College. Victoria Institute. Zoological Society. Manchester. AB. Free Library. AB. Literary and Philosophical Society. p. Geological Society. AB. Owens College. Netley. p. Royal Victoria Hospital. Newcastle. Free Library. North of England Institute of Mining and Mechanical Engineers. p. Society of Chemical Industry (Newcastle Section) . Norwich. p, Norfolk and Norwich Literary Institution. Nottingham. AB. Fi'ee Public Library. A. P' P' V' V- p- p- A. B. B. AB. P' P- P^ AB. B. P' P' P- AB. AB. P- P' AB. AB. AB. B. AB. P' England and Wales ^continued). Oxford. p, Ashmolean Society. AB. Radcliffe Library. A. Radcliffe Observatory. Penzance. p. Geological Society of Cornwall. Plymouth. B. Marine Biological Aflsociation. p. Plymouth Institution. Richmond. A. " Kew " Observatory. Salford. p. Royal Museum and Library. Stonyhurst. p. The College. Swansea. AB. Royal Institution. Woolwich. AB. Royal Artillery Library. Finland. Helsingfors. p. Societas pro Fauna et Flora Fennica. AB. Soci^te des Sciences. France. Bordeaux. P- V' P- P- Academic des Sciences. Faculte des Sciences. Soci6t^ de M6decine et de Chirurgie. Soci^t^ des Sciences Physiques et Naturelles. Caen. p. Soci^t^ Linn6enne de Normandie. Cherbourg. p. Soci6te des Sciences Naturelles. Dijon. p. Academic des Sciences. Lille. p. Faculty des Sciences. Lyons. AB. Academic des Sciences, Belles - Let tres et Arts. AB. University. Marseilles. AB. Faculte des Sciences. Montpellier. AB. Academic des Sciences et Lettres. B. Faculty de M6decine. Nantes. P- Paris. AB. P- P' Soci^te des Sciences Naturelles de TOuest de la France. Academic des Sciences de Tlnstitut. Association Fran9ai8e pour TAvancement des Sciences. Bureau des Longitudes. h 2 Digitized by Google [ xvi ] Qnited States (continued). Philadelphia. AB. Academy of Natural Sciences. AB. American Philosophical Society. p. Franklin Institute. p. Wagner Free Institute of Science. Rochester (N.Y.). p. Academy of Science. St. Louis. p. Academy* of Science. Salem (Mass.). p. American Association for the Adyance- ment of Science. AB. Essex Institute. San Francisco. AB. California Academy of Sciences. United States (continued). Washington. AB. Patent Office. Smithsonian Institution. United States Coast Surrey. United States Commission of Fish and Fisheries. United States Geological Survey. United States Nayal Observatory. United States Department of Agriculture. United States Department of Agriculture (Weather Bureau). West Point (N.Y.) AB. United States Military Academy. AB. AB. B. AB. AB. P' A. Digitized by Google I xvii ] Adjudication of the Medals of the Royal Society for the year 1900, by the President and Council. The COPLEY MEDAL to Professor Marcellin Berthelot, For. Mem.R.S., for his brilliant services to Chemical Science. The RUMFORD MEDAL to Professor Antoine Henri Becquerel, for his discoveries in Radiation proceeding from Uranium. A ROYAL MEDAL to Major Percy Alexander MacMahon, F.R.S., for the number and range of his contributions to Mathematical Science. A ROYAL MEDAL to Professor Alfred Newton, F.R.S., for his eminent contributions to the science of Ornithology and the Geographical Distribution of Animals. The DAVY MEDAL to Professor Guglielmo Koerner, for his brilliant investi- gations on the Position Theory of the Aromatic Compounds. The DARWIN MEDAL to Professor Ernst Haeckel, for his long-continued and highly-important work in Zoology, all of which has been inspired by the spirit of Darwinism. The Bakerian Lecture for the year 1900, " On the Specific Heat of Metals and the Relation of Specific Heat to Atomic Weight," was delivered by Professor W. A. Tilden, F.R.S., on March 8, 1900. The Croonian Lecture for the year 1900, " On Immunity with Special Reference to Cell Life," was delivered by Professor Dr. Paul Ehrlich, on March 22, 1900. VOL. CXCV. — A. C Digitized by VnOOQ iC Digitized by VjOOQ IC PHILOSOPHICAL TRANSACTIONS. I. Mathematical Contributions to the Theoiy of Evolution. — VII. On the Correlation of Characters not Quantitatively Measurable. By Karl Pearson, F.R.S. {From tlie Deparimeitt of Applied Mathematics^ University College^ London,) Eeceived February 7,— Read March 1, 1900. NOTE. In August, 1899, I presented a memoir to the Boyal Society on the inheritance of coat-colour in the horse and of eye-colour in man, which was read November, 1899, and ultimately ordered to be published in the * Phil. Trans.' Before that memoir was printed, Mr. Yule's valuable memoir on Association was read, and, further, Mr. Lesue Brahley-Moore showed me that the theory of my memoir as given in § 6 of the present memoir led to somewhat divergent results according to the methods of proportioning adopted. We therefore undertook a new investigation of the theory of the whole subject, which is embodied in the present memoir. The data involved in the paper on coat-colour in horses and eye-colour in man have all been recalculated, and that paper is nearly ready for presentation.^ But it seemed best to separate the purely theoretical considerations from their application to special cases of inheritance, and accordingly the old memoir now reappears in two sections. The theory discussed in this paper was, further, the basis of a paper on the Law of Eeversion with special reference to the Inheritance of Coat-colour in Basset Hoimds recently communicated to the Society, and about to appear in the * Proceedings.*! While I am responsible for the general outlines of the present paper, the rough draft of it was taken up and carried on in leisure moments by Mr. Leslie Bramley-Moore, Mr. L. N. G. Filon, M.A., and Miss Alice Lee, D.Sc. Mr. Bramley-Moore discovered the w-functions ; Mr. Filon proved most of their general properties and the convergency of the series ; I alone am responsible for sections 4, 5, and 6. Mr. Leslie Bramley-Moore sent me, without proof, on the eve of his departure for the Cape, the general expansion for z on p. 26. I am responsible for the present proof and its applications. To Dr. Alice Lee we owe most of the illustrations and the table on p. 17. Thus the work is essentially a joint memoir in which we have equal part, and the use of the first personal pronoun is due to the fact that the material had to be put together and thrown into form by one of our number, — K P. Contents. page § 1. On a General Theorem in Normal Correlation for two Variables. Series to Determine the Correlation 2 § 2. Other Series for the Determination of the Correlation 7 * Since ordered to be printed in the * Phil. Trans.' t Bead January 25, 1900. ' Roy. Soc. Proc.,' vol. 66, p. 140. VOL. CXCV.— A 262. B 16.8.1900. Digitized by Google 2 PROFESSOR K. PEARSON ON iMATHEMATICAL CONTRIBUTIONS § 3. Proof of the General Convergency of the Series for the Correlation 10 § 4. On the Probable Error of the Correlation thus Determined 10 § 5. To Determine a Physical Meaning for the Series and on Divers Measures of Association ... 14 §6. On the " Excess " and its Relation to Correlation and Relative Variability 18 § 7. On a Generalisation of the Fundamental Theorem of the present Memoir. Special Formulae for Triple and Quadruple Correlation 23 § 8. Illustrations of the Methods of the Memoir 35 Illustration I. Inheritance of Coat-colour in Thoroughbred Horses. Sire and Filly 35 „ II. Chance that an Exceptional Man is bom of an Exceptional Father 37 „ III. Inheritance of Coat-colour in Dogs, Half-" Siblings " 38 „ IV. Inheritance of Eye-colour between Maternal Grandmother and Granddaughter . . 39 „ V. Inheritance of Statiu^e between Father and Son for different groupings 40 „ VI. Correlation between Strength to resist Small-pox and Degree of Effective Vaccination 43 „ VII. Effect of Antitoxin on Diphtheria Mortality 44 „ VIII. Chance of Stock above the Average giving Produce above the Average as compared with the chance of such Produce from Stock below the Average 45 „ IX. Chance of an Exceptional Man being bom of Exceptional Parents 46 § (1.) On a General Theorem in Normal Correlation. Let the frequency surface N z = where 27rv/(l - r^)<ri<r^ N = total number of observations, cTi, 0*2 = standard deviations of organs x and y, r = correlation of x and y, be divided into four parts by two planes at right angles to the axes of x and y at distances h' and k' from the origin. The total volumes or frequencies in these parts will be represented by a, 6, c, and d in the manner indicated in the accompanymg plan : — TdiJb/e of Frequenc/ea dL b <SL'¥b C d c^d d*C b^d N Then clearly c? = 27rv/(l ^==- f fe- »i473(«' + 1" - ^^y) dxdy. . (1 - r»)J» Jt if 27rv/(i - r») . h = h'/a-i and k = h'/tr^ (i.). Digitized by Google Further, and TO THE THEORY OF EVOLUTION, N_ r _ *^(fo; ("•). ^ V TT Jo <-^^ti)^^^=vif:e-% ■. . . .w. Thus, when a, 6, c, and dl are known, h and ifc can be found by the ordinary table of the probability integral, say that -of Mr. Sheppard (* Phil. Trans.,' A, vol. 192. p. 167, Table VI.*). The limits accordingly of the integral for d in (i.) are known. Now consider the expression .e-*rr^<*' + ^-^'^> = U,say, (vi.), x/1 - r« and let us expand it in powers of r. Then, if the expansion be /'d»U\ (vii.). (viii.). we shall have Taking logarithmic differentials, we get at once (1 _ r^)2^ = {xy + r(l - a^ - y^) + i^xy - 7^}JJ. Differentiating n times by Leibnitz's theorem, and putting r = 0, we have, after some reductions ^«+i = ^(2n — 1 — x^ — y*)w^i - n{n —l){n — 2fun.^ + ^{un + n{n — l)Un^^} (ix.). Hence we find Wo = = 1 Mi = -xy «s = = («»- •1)(3^-1) ^3 = = a;(a:»- -3)y(y«-3) W^r = («*- •6a:^ + 3)(y*-. 6y« + 3) J * See, however, foot-note , p. 5. B 2 (X.) Digitized by VjOOQ IC 4 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Thus the following laws are indicated : — Un = v„X w„ (xL), where v, = an;,_i — (n — l)v<,_g (xiL), ■w« = yw^-i — (« — l)«'«-9 (xiii.). We shall now show that these laws hold good by induction. . Assume Thus M,+i = xyu„ + n^u^_i — n(yw,v._i + aw,u>._j). But by (ix.), substituting for m,_3 from (xi.) and (xiii.), ««+i = ay {v„w„ + w(n — 1)v,_2m;,_2J + n{2n —l—x^ — i/*)i;«_itp,_, — n{n — l)t\_iw,_i — ycyn{n — l)v„_8w;,_3 + n(n — 1) (yt',_iu»_2 + xVn^^Wn-i). 4- n(n — 1) (y v„_iW,_s + a5t;,_gw,_i) + a;u>,_i(a;v,_i — n— lv«_8} = v»+iW,+i, as we have seen above. Thus, if the theorem holds for u^, it holds for m,+i. Accordingly where the 7/s and v/b are given by (x.), (xiL), and (xiii.). It is thus clear that k~ \ I V dx dy consists of a series of which the general term is 1 n V«W«r* 1 f* where V„ = -y--= e"**'v„da; '^'''■^\y-^«-<^- It remains to find these integrals. The general form of v^ is given by .. ==af-'^%^-^af-^+ "<- - '^%-, ^)<" - ^> af-* - &c. . . (xv.). Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 5 For this obviously gives (x.). Assume it true for v«_i and v„_2, then — XT 2U ^ + 22|2^ * — . . . = «». Thus the expression (xv.) is shown to hold by induction, the general terms being or the general term in u^. We notice at once that dx ^r = ^^«-i (^^0- Thus, by (xii.) dx Vn — XVn^^ — Multiply by e"*"^ and integrate Integrating the latter integral by parts, we have Now y— e"*** can be found from any table of the ordinates of the normal curve, e.g., Mr. Sheppakd's, * Phil. Trans.,' A, vol. 192, p. 153, Table I.* We shall accord- ingly put = = C4«-'" K=^eH' (xvu.), and look upon H and K as known quantities. * For our present purposes the differences of Mr. Sueppa.rd's tables are occasionally too large, but the following series give very close results : — Let '^^ = Vl (<.-«• c)- (6 + d) ^ I V,^ ^ ^y (j^ )^ Digitized by VjOOQ IC 6 PEOFESSOR K. PEAESON ON MATHEMATICAL C50NTRIBUTI0NS Further, let us write (v«-i)* » * as v«. j, and similarly (M;«_j)y » * as Ww.,. Thus V, = H.'U«.„ W, = K.'M>,_i (xviil). We have then from (i.) = ill \y''''''<^dy + s(|hk^,.,^._,) (ft + d)(<! + rf) "/r" - - \ by (ii.) and (iii.). Or, remembering that N = a + 6 + c4-c?>we can write this ad — he 2 /r"- - \ = r + ^ M + J (A« - 1) (i« - 1) + gA(A« - 3)*(i« -3) + r^(A« - 15¥ + 45^2 - 15) (Jfc» - 15;fc* + 45*2 _ 15) 5040' 40320' + ifti^A(A« - 21^* + 105A« - 105)ife(P - 21^ + 105*^- 105) + , &c. . . . . (xix,). Then A = xi + |lx.« + ^Xi' + ^Xi' + and 1 ,75--/, 1 2.7 . , 127 . , H = ^^"i^ ■*• [2;^^ + [4 >^^ + 76->^^ + ■ 1 7 127 * = X2 + j3 X2» + jg- X2* + -ry-X2^ + • • • 1 Ts-/, 1 Ys 7 . 127 . These follow from the considerations that if Xi = V2«-<^. Xj = V2t*s. d<h „ t=K. iH . #1 == - *' whence it is easy to find the successive differentials of h with regard to <f>i and k with regard to <^2> ^nd then obtain the above residts by Maclaurin's theorem. There is, of course, no difficulty in calculating % H and K from (xvii.) directly. That method was adopted in the niunerical illustrations. j Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 7 Here the left-hand side is known, and since h and k are known, we can find the coefficients of any number of powers of r so soon as the first two have been found, firom (xii.) and (xiii.). Accordingly the correlation can be found if we have only made a grouping of our frequencies into the four divisions, a, 6, c, and d. If h and k be zero, we have fi:om (xvii.) and (iv.) H = K= ^ i/27r The right-hand side of (xix.) is now ^ . c - - it? ^ ^ r+f3r» + 5o <x ■ (L a**^ t - K,. or equal to sin"^ r. _. . ^ (ad — 6c) Hence r — sm 2ir ^^ — rr^ — = cos IT —2 (XX.), which agrees with a result of Mr. Sheppard's, * Phil. Trans.,' A, voL 192, p. 141. We have accordingly reached a generalised form of his result for any class-index whatever. Clearly, also, r being known, we can at once calculate the fi:equency of pairs of organs with deviations as great as or greater than h and k. § (2.) Other Series for the Determination of r. For many purposes the series (xix.) is sufficiently convergent to give r for given h and k with but few approximations, but we will now turn to other developments. We have by (vii.) Put X = hf y = k, and write for brevity ad ^ be c = N«HK It follows at once from (xix.) that (xxL). dr ^^0 Digitized by VjOOQ IC 8 PKOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS — qW g-i(it tan « - A«ec «)« ^^ if r = sin 6. Now either of the quantities under the sign of integration in (xxii.) can be expanded in powers of 6 by Maclaurin's theorem. Thus let = Xo + (g)/ + (^)|+...+(g)i + ... Then and it remains to find ( -tx I • Now log X = — i (^ tan ^ — A sec dy. Hence cos* ^ ^ = - X [(A' + ^•*) sin ^ _ M (A - ^ cos 2^)]. Differentiating n — 1 times by Leibnitz's theorem, and putting ^ = 0, Clearly Xo = ^"**'> then we rapidly find Or, finally c = ^ + 1M^2 - (A^ + P - A^P) ^ + ;i&{;i.2ifc» - 3(A^ + p) + 5} ^ + . . . (xxiv.), where more terms if required can be found by (xxiii.). If 6 be fairly small, 6^ will be negligible. Or if A and k be small, the lowest terra in the next factor will be h^ -f /:^, Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 9 and this into ^/|5 is generally quite insensible/ Very often two or three terms on the right-hand side of (xxiv.) give quite close enough values of ^, and accordingly of r = sin^. (xxiv.) is clearly somewhat more convergent than (xix.) if ^ and h are, as usually happens, less than unity. Returning now to (xix.), let us write it e=f{r,h,k). This is the equation that must be solved for r. Suppose r^ a root of this when we retain only few terms on the right, say a root of the quadratic € = r + p.*r*. Then if r = Tq + /»> « =/(ro, h, k) + pf{r^, h, k) + \ipY"{r„ h, k) + &c. Hence p = ^ ./ vH to a third approximation V^-r, 1 ir:^^;^;^-^;^^ nearly (XXV.), which gives us a value of p which, substituted in p^ in the above equation, introduces only terms of the 6th order in r,^. Another integral expression for c of Equation (xxi.) may here be noticed : )v/l PutA=-^(^ + y),A: = -^(^-.y) Hence Jo\/l — ^ Jo Vi — T* 1 — r Let tan 2^ = , or, r = co82 if>. Therefore f46<» Jl 1 +V^ where t; = cot and is > 1. It seems possible that interesting developments for € might be deduced from this integral expression. VOL. cxcv.— A. o Digitized by VnOOQ iC 10 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS § (3.) To show that the SeiHesfor r is Convergent i/ r < 1, whatever be the Values of h and k Write the series in the form of p. 6, i.e. : — Now W,+i = Aw. — WW,_i J •' ^ ' From these we deduce V»+l = {^' — (2W — 1)} Vn-i — (n — 1) (w — 2) v,_3 ^«+i = {** — (2w — 1)} w,_i — (n — 1) (n — 2) w,_3 Now let «» = iLi»**|{|»}*, «« =~M'»-i»'*"| {!«}*. Then we find ^'•+* "" v/(n+l)(7. + 2) ^-"^ V 71(71+1) (71 +2; ^'^^'^ ' ^-■^^ - ^(n + l)(7i + 2) ^-^ V n(7i + l)(ri + 2) ^'-^"^ ' Thus, when 7^ is large, we find the ratio of successive terms Sn^Js^ or t^+cj^n is given by /o, where /) = — 2r — r^//o or, /o = — r. The ultimate ratio of s^+2 '*»+2 *^ ^» '« ^s accordingly given by r*, but this is the ratio of alternate terms of the original series. The original series thus breaks up into two series, one of odd and one of even powers of r. Both these series are absolutely convergent whatever h and k be, having an ultimate convergence ratio of r ^ § (4.) To find the Probable Error of the Coirelation Coefficient as Determined by the Method of this Memoir. Given a division of the total frequency N into a, 6, c, d groups, where a + & + c + d = N, then the probable error of any one of them, say a, is '67449 cTa, where* <r«= y/^^ (xxvi.). Let 6 + c? = nj, c + d = 7^2, then * The standard deviation of an event which happens np times and fails nq times in n trials is well known to be Jnpq. The probable errors here dealt with are throughout, of course, those arising from different samples of the same general population. Digitized by VnOOQ iC TO THE THEORY OP EVOLUTION. 11 --. = a/^^^ <r. = ^/^^ . . . (xxvil). To obtain Vcd we have, if Br/ denotes an error in any quantity tj, Sc + 8d = Sng, .-. a-J^ + <r/ + 2a'ca'drcd = <r«.® (xxviii.), by squaring, summing for all possible variations in c and rf, and dividing by the total number of variations. Hence, substituting the values of the standard deviations as foxmd above, we deduce o-cO-dTed = — ccZ/N (xxix.). In a similar manner hn^U=hhU + (U)\ O'dO'n^rdn^ = O-hO-dTbd + Ct/ o-rfo-^^r^^ = d (a + c)/N (xxx.). and <rrfO-„,ri/,^= ci (a + &)/N (xxxi.), N f* Now ^1 ~ "TT"] ^"^^^ Thus (r^=NH(rA (xxxiL), and similarly Cn^ = NK o-^ . . (xxxiii.). Hence the probable error of h =^ ^25^13 (^,w.), andofi =|^^<I±^±I). ..... .(X.ZV.). They can be found at once, therefore, when H and K have been found from an ordinate table of the exponential durve, and a, 6, c, d are given. We have thus the probable error of the means as found from any double grouping of observations. Next, noting that 8«iK = N*HK8A8Jfc, we have (r„, <^njrn,H^ — N^HK Ck o-kr^, or Tn^n, = Tkh. c 2 Digitized by VnOOQ iC 12 PEOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS But 8ni 8% = (S6 + Sd) (8c + 8d), = j^ . (xxxyi.), therefore <Tk(Tkrhk— ygj^ (xxxvil). ad — he , ... . ^^~ v/(T+rf)(aH-c)(c + ei)(a + 6) • • • • • (xxxvui.). This is an important result ; it expresses the correlation between errors in the position of the means of the two characters under consideration. But if the prob- abilities were independent there could be no such correlation. Thus r;^ might be taken as a measure of divergence from independent variation. We shall return to this point later. Since S^i = — HNSA, we have Sn^Sci = — HNSrfS^, whence we easily deduce rdn,— --rah (xxxix.). Similarly ^if«,= '^'^dk • . (xl.). Now d is a function of r, h^ and k. Hence if d = f{r^ A, k)^ 8rf=^8r + f 8^ + |8ik dr dh ok = 70^^ + 71^^ + 72^* (xli.)- Whence transposing, squaring, summing, and dividing by the total number of observations, we find yo^o-r^ = cr/ + y^o-^ + y^o-j?' — ^y^a-dO-krdh — ^y^o-dO-kTdk + "^yiy^o-ho-kThk Substituting the values of the standard deviations and correlations as found above, we have V» = j^ { c/(a + 6 + c) + (^^(a + h) {d + c) + (^)'(a + c) (d + b) Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 13 It remains now to determine yo, yi, and y^. By Equation (i.) ' _df N = -iiJ;l>-"^^ (-!-)> where A= ^7-r« - Thus = '/'2-i (xlv.). Sinailarly •/^/(NK) = ^^ — ^ . . .- ^ . . . . (xlvi.). H^^^ ^^ = ;7feD"*'^^' ^^=:^D"''^- • • • -(^^^-^^ where A= yfJ^ > ^^ = TT^ (xlviii.), and thus t/r^ and i/r^ can be found at once from the tables when fi^ and ^82 are found from the known values of r, h^ k. Lastly, we have from Equation (xxi.) d (d + h)(d + e) . 1 f' TJ Thus* y^ = d//dr = ~l^U, ro/N = xo. where Xo = ^ -^i-^ e-.ir^ <»» + **-*") (xlix.) a value which can again be foiind as soon as r, ^, A; are known, y^ s ^^ is clearly the ordinate of the frequency surface corresponding to x =^ h,y ^= L Substituting in Equation (xliiL) we have, after some reductions, * By Equations (ii.) and (iii), d + b and d + e are independent of r. Digitized by VjOO^ IC 14 PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS Probable error of r s= •67449<rr •67449 r (a + d)(e+ b) ^ (a + c)(d + b) ^ (<^ + b)(d+e) y/K^ \ 4N« "^ '''» N» "^ '''1 N» ad-^hc 'ah — ed , ac — 6d 1 i N» (1). where Xo> ^i> ^^^ ^a ^^^ readily found from Equations (xlix.), (xlvii.), and (xlviii.). ' Thus the probable error of r can be fairly readily found. It must be noted in using this formula, that a is the quadrant in which the mean falls, so that h and k are both positive (see fig., p. 2). In other words, we have supposed a + c > 6 + r? and a + 6 > c + d. Our lettering must always be arranged so as to suit this result j before we apply the above formula. § (5.) To Find a Physical Meaning for the Series in ?% or for the € of Equation (xxL). I h Return to the original distribution ^ , of p. 2. If the probabilities of the two characters or organs were quite independent, we should expect the distribution N a-\-l a •{- c N N N c •{- d a-i- c N g -f ft b '\' d N N N c-h db + d N N N N Now re-arranging our actual data we may put it thus a -i- b b + d a I b N N N N e + d a + c ad — be N (N N ad— be n: N N ad --be T^T-c + dft + d , orf — 6c N N N Accordingly correlation denotes that — jx— has been transferred from each of the second and fourth compartments, and the same amount added to each of the first and third compartments. If t^ = {ad — 6c)/N^ then ri is the transfer per unit of the total frequency. The magnitude of this transfer is clearly a measure of the divergence of the statistics from independent variation. It is physically quite as significant as the correlation coeflEicient itself, and of course much easier to determine. It must vanish with the correlation coefficient. We see from (xxi.) that ri = €X HK, or we have an interpretation for the series in r of (xix.). Now, obviously any function of 17, just like ri itself, would serve as a measure 01 the divergence from perfectly independent variation. It is convenient to choose a function which shall lie arithmetically between and !• Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 15 Now consider what happens in the case of perfect correlation, i.e., all the observa- tions fall into a straight line. Hence if ad > 6c, either 6 or c is zero, for a straight line cannot cut all four compartments, and a and d are obviously positive. Thus c and 6 can only be zero if i/ = (c + d){a + c)/N^ or (a + h){b + d)fN\ In order that 6 should be zero, it is needful that h and A, as given by (iv.) and (v.), should be positive ora+c>6+d, a + 6>c + c?, and the mean fall under the 45° line through the vertical and horizontal lines dividing the table into four compartments, i.e.y h > k. These conditions would be satisfied if ad >hc and a > d^ c > b. Now suppose our four-compartment table arranged so that ad>hCy a>d, c>by and consider the function Q' = ^^^f(a + 6)(5 + cO/N' (^•)' or ^^ . TT ad — be ,,,. V Q' = "^"2 (a.f&)(5 + rf) <^->- This ftmction vanishes if 17 = 0, and it further = unity if 6 = 0. Thus it agrees at the limits and 1 with the value of the correlation coefficient. Again, when h and k are both zero, a = d, 6 = c, and Q^ = sin n- ^ . ^ , is thus r by (xx.). Hence we have found a ftinction which vanishes with r and equals unity with r, while it is also equal to r if the divisions of the table be taken through the medians. Now, I take it that these are very good conditions to make for any function ot" a, 6, c, d which is to vanish with the " transfer," and to serve as a measure of the degree of dependent variability, or what Mr. Yule has termed the degree of ** association." Mr. Yule has selected for his coefficient of association the expression QCUL ""^ OC /« . • • * This vanishes with the transfer, equals unity if 6 or c be zero, and minus unity if a or (i be zero. The latter is, of course, unnecessary if we agree to arrange a, 6, c, d so that ad is always greater than be. Now it is clear that Q2 possesses a great advantage over Q^ in rapidity of calculation, but the coefficient of correlation is also a coefficient which measures the association, and it is a great advantage to select one which agrees to the closest extent with the correlation, for then it enables us to determine other important features of the system. If we do not make all the above conditions, we easily obtain a number of coeffi- cients which woyld vanish with the transfer. Thus for example the correlation Vkjt of Equation (xxxviii.) is such an expression.* It has the advantage of a synmietrical . form, and has a concise physical meaning. It does not, however, become .imity when * In fact (xxxvii) gives us c » a-jiO'tirkk' Digitized by Google 16 PROFESSOR K, PEARSON ON MATHEMATICAL CONTRIBUTIONS either, but not both, 6 and c vanish, nor does it, unless we multiply it by 7r/2 and take its sine, equal the coefficient of correlation when a ^=^ d and b = c. Again, we might deduce a fairly simple approximation to the coefficient of correla- tion from the Equation (xxiv.) for ^, using only its first few terms. Thus we find f>*. dd — he /!• \ 8m 27r ^,^^ _ ^^^, ^ ^^.^^ ^ ^^^^^^^ ^ ^^ (liv.), where Xi = V 2 N ' X^ "" V 2 N ' as an expression which vanishes with the transfer, and will be fairly close to the coefficient of correlation. It is not, however, exactly unity when either 6 or c is zero. But without entering into a discussion of such expressions, we can write several down which fully satisfy the three conditions : — (i.) Vanishing with the transfer. (ii.) Being equal to unity if 6 or c = 0. (iii.) Being equal to the correlation for median divisions. Such are, for example : — Q« = smjf:j : 26c ^ ,od>}>c . . . (Ivi.), {ad — be) (b + c) where i^ = 2 ^1 Aahcd W (ad-'bcy(a-i-d)(b + c) Only by actual examination of the numerical results has it seemed possible to pick out the most efficient of these coefficients. Q^ was found of little service. The following table gives the values of Qg, Q3, Q4, and Q5 in the case of fifteen series selected to cover a fairly wide range of values : — Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 17 No. r. h. k. Q2. Qs. 04. 0*. 1 •5939 ± -0247 - •0873 - ^4163 •7067 •6054 •6168 •6100 2 •5557 ± -0261 - ^4189 - 4163 •6688 •5657 •5405 •5570 3 •5529 ± ^0247 - •0873 - ^0012 •6828 •5809 •5699 •5813 . 4 •5264 ± -0264 + ^2743 + •3537 •6345 •5331 •5200 •5283 5 •5213 ± ^0294 + ^6413 + ^6966 •6530 •5511 •4878 • •5160 6 •5524 ± -0307 + r0234 + 3537 •7130 •6118 •6169 •6138, 7 •5422 ± -0288 + ^6463 + ^5828 •6693 •5673 •5136 •5452 8 •2222 ± •0162 + ^3190 + ^3190 •2840 •2268 •2164 •2251 9 •3180 ± ^0361 + ^1381 + ^0696 •3959 •3185 •3176 •3183 10 •5954 ± ^0272 + 1^5114 + •7414 •7860 •7100 •6099 •6803 - 11 •4708 ± -0292 + ^0865 - -0054 •5692 •4712 •4720 •4715 12 •2335 ± •0335 + ^0405 + ^0054 •2996 •2385 •2385 •2385 13 •2451 ± ^0205 + -2707 + ^0873 •3103 •2473 •2456 •2470 14 •1002 ± ^0394 + -4557 + ^1758 •1311 •1032 •0993 •1029 15 •6928 ± ^0164 + -5814 + -5814 •8032 •7108 •6699 •6897 Now an examination of this table shows that notwithstanding the extreme ele- gance and simplicity of Mr. Yule's coefficient of association Q^, the coefficients Q3, Qt, and Q5, which satisfy also his requirements, are much nearer to the values assumed by the correlation. I take this to be such great gain that it more than counterbalances the somewhat greater labour of calculation. If we except cases (6) and (10), in which h or k take a large value exceeding unity, we find that Q3, Q^, and Qg in the fifteen cases hardly differ by as much as the probable error from the value of the correlation. If we take the mean percentage error of the difference between the correlation and these coefficients, we find Mean difference of Q^ = 24*38 per cent. » >> ^3 ^^ 3'9o ,, „ Q^= 2-94 ,, Q5= 272 Thus although there is not much to choose between Q^ and Q5, we can take Q5 as a good measure of the degree of independent variation. The reader may ask : Why is it needful to seek for such a measure ? Why cannot we always use the correlation as determined by the method of this paper ? The answer is twofold. We want first to save the labour of calculating r for cases where the data are comparatively poor, and so reaching a fairly approximate result rapidly. But labour-saving is never a wholly satisfactory excuse for adopting an inferior method. The second and chief reason for seeking such a coefficient as Q lies in the fact that all our reasoning in this paper is based upon the normality of the frequency. We require to free ourselves from this assumption if possible, for the difficulty, as is exemplified in Illustration V. below, is to find material which actually obeys within the probable errors any such law. Now, by considering the coefficient of regression, ra-Jc^ = 8{xy)/Q^a'ia'^), as the slope of the line which best fits the series VOL. CXCV, — ^A. D Digitized by Google 1 8 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS of points determined as the means of arrays of x for given values of y, we have once and for all freed ourselves from the difficulties attendant upon assuming normal frequency. We become indifferent to the deviations from that law, merely observing how closely or not our means of arrays fall on a line. When we are not given arrays but gross grouping under certain divisions, we have seen that the " transfer " is also a physical quantity of a significance independent of normality. We want accordingly to take a function which vanishes with the transfer, and does not diverge widely from the correlation in cases that we can test. Here the correlation is not taken as something peculiar to normal distributions, but something significa^t for all distribu- tions whatever. Such a function of a suitable kind appears to be given by Qg. § 6. On the " Excess " and its Relation to Correlation and Relative Variability. There is another method of dealing with the correlation of characters for which we cannot directly discover a quantitative scale which deserves consideration. It is capable of fairly wide application, but, unlike the methods previously discussed, it requires the data to be collected in a special manner. It has the advantage of not applying only to the normal surface of frequency, but to any surface which can be converted into a surface of revolution by a slide and two stretches. It is well known that not only the normal curve but the normal surface has a type form from which all others can be deduced by stretching or stretching and sliding. Thus in 1895 the Cambridge Instrument Company made for the instrument room at University College, London, a " biprojector," an instrument for giving arbitrary stretches in two directions at right angles to any curve. In this manner by the use of type-templates we were able to draw a variety of curves with arbi- trary parameters, c.gr., all ellipses from one circle, parabolas from one parabola, normal curves from one normal curve template. Somewhat later Mr. G. U. Yule commenced a model of a normal frequency surface on the Brill system of inter- laced curves. This, by the variable amount of slide given to its two rectangular systems of normal curves, illustrated the changes from zero to perfect correlation. This model was exhibited at a College soirSe in June, 1897. Greometrically this property has been taken by Mr. W. F. Sheppard as the basis of his valuable paper on correlation in the *Phil. Trans.,' A, vol. 192, pp. 101-167. It is a slight addition to, and modification of, his results that I propose to consider in this section. The equation to the normal frequency surface is, as we have seen in § 1, _ N r /^ 2rxy , y^\ 1 ] Now write a;/(<riN/l— r^) = x\ y/c^ = y\ This is merely giving the surface two uniform stretches (or squeezes) parallel to the coordinate axes. We have for the frequency of pairs lying between x^ x + Sx, and y, 8 + 8//, Digitized by Google TO THE THEORY OF EVOLUTION. 19 Now give the area a uniform slide parallel to the axis of x defined by r/y/l— r* at unit distance from that axis. This will not change the basal unit of area 8a = 8x'By\ and analytically we may write Whence we find ^ ^ N ^ , -ti^v zoxoy = 2^ 8a expt. (— ^^K*). This is the mechanical changing of the Yule-Brill model analytically represented. The surface is now one of revolution, and the proof would have been precisely the same if we had written in the above results any function/, instead of the expo- nential.* It is easy to see that any volume cut off by two planes through the ^is of the surface is to the whole volume as the angle between the two planes is to four right angles. Further the corresponding volumes of this surface and the original surface are to each other as unity to the product of the two stretches. Lastly, any plane through the z-axis of the original solid remains a plane through the 2:-axis after the two stretches and the slide. These points have all been dealt with by Mr. Sheppard (p. 101 et seq.y loc. cit.). I will here adopt his notation r = cosD, and term with him D the divergence. Thus cot D is (in the language of the theory of strain) the slide, and D is the angle between the strained positions of the original x and y directions. Now consider any plane which makes an angle x with the plane of xz before strain. Then, since the contour lines of the correlation surface are ellipses, the volumes of the surface upon the like shaded opposite angles of the plan diagram below will be equal ; and if they be n^ and n^, then n^ + n^ = ^N. If n{ and n^' be the volimies after strain, then by what precedes we shall have and (rig - ni)/(ni + n^) = « - <)/« + <)• -a? 8 * The generalisation is not so great as might at first appear, for I have convinced myself that this property of conversion into a surface of revolution by stretches and slides does not hold for actual cases of markedly skew correlation. d2 Digitized by Google 20 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Now Ui and n^' will be as the angles between the strained positions of the planes bounding n^ and n2* ^^ Aoes not change its direction. Oy is turned through an angle 7r/2 — D clockwise, and x becomes x\ say. Hence <:<::|-x"+|-I>:l + x"-I+D. or (V - n^)/(n^ + n/) = | (x" + D) - 1. Let us write E^ = 2(% — n^ and term it the excess for the y-character for the line AB. Then we easily find : /E, TT , 7r\ . / // . -rw\ cot y' + cot D ,1 ... . It remains to determine tan ^ ' and substitute. The stretches alter tan x into tan x', such that tan X = ~^ tan x- Further, by the slide cot y" = cot y' — cot D = — ,-^ - -^ cot Y — cot D. Hence we have by (Iviii.) above - -* (s !) = i.-;/Tr? -t x/(;;7fci oot X oot D - cof D - l) , -tan(|jUootD-a55^ (li^). \N 2/ (Tg smD ^ ^ or. Now the excess E^ is the difference of the frequencies in the sum of the strips of the volume made by planes parallel to the plane yz on the two sides of the plane AB2; (defined by x)> taken without regard to sign. For on one side of the mean yy this is ri2 — n^y and on the other — (^i— ^2)- Hence we have this definition of E^, the column excess for any line through the mean of a correlation table : Add up the frequencies above and below the line in each column and take their differences ivithout regard to sign^ and their sum is the column excess. If we are dealing with an actual correlation table and not with a method of collecting statistics, then care must be taken to properly proportion the frequencies in the colunm in which the mean occurs, and also in the groups which are crossed by the line. It is the difficulty of doing this satisfactorily, especially if the grouping, as in eye and coat colour, is large and somewhat rough, that hinders the effective use of the method, if the statistics have not been collected ad hoc. Now let E2 be the row excess for the line AB, defined in like manner, then we have in the same way Digitized by VnOOQ iC TO THE THEORY OP EVOLUTION. 21 -tan(|^j)=:cotD-^»^ (Ux>). \N 2/ a-j^ smD ^ ^ Now eliminate o-^/cti between (lix.) and (lix.^") ; then (tan (I I) + oot D) (tan (I f ) + cot D) = „^ . Whence we deduce and, therefore, cot D = cot ^^ ~, N 2 DEj + Eg TT ' n \ = COS ^ 2 V^-)* Substituting for D in (lix.) we find further ^; = cotxcos(||yco8(|^) (IxL). Thus Equations (Ix.) and (Ixi.) give the coefficient of correlation and the relative variability of the two characters. The latter is, I believe, quite new, the former novel in form. If we call nil ^^® frequency in the angle x {-^Ox of the figure above), then it is easy to see that E^ = 2(^2 — n^) = N — 471^, and similarly E^ = N — 4m|. Thus (El + E2)/N = 2(N — 2(ni + mi))/N. But n^ + m^ is the frequency in the first quadrant. This Mr. Sheppard terms P, while that in the second he terms R. We have thus (E^ + E2)/N = 2ll/(Il + P), or p ^ = cos^-^7r (Ixii.), ie.y Mr. Sheppard's fimdamental result* (* Phil. Trans.,' A, vol. 192, p. 141). We can, of course, get Mr. Sheppard's result directly if we put x = 0, when we have at once E^ = 2(R — P), Eg = N :;= 2(R + P), and the result follows. Equation (Ixi.) may also be written in the form ^ = cotxsin(^'2^)/8in(^2^) (Ixiii.). If we put x^ ^y t^^^ ^1 becomes zero, and the right-hand side of (Ixiii.) is indeterminate. If we proceed, however, to the limit by evaluating the frequency in an indefinitely thin wedge of angle Xj we reach merely the identity cja-^ = o-i/o-^. Hence there is no result corresponding to (Ixi.) to be obtained by taking Mr. Sheppard's case of x = 0. The following are the values of the probable errors of the quantities involved : — * In the actual classification of data (Ix.) and (Ixii.) suggest quite different processes. We can apply (Ix.) wbere (Ixii.) is difficult or impossible, e.g., correlation in shading of birds' eggs from the same clutch. Digitized by VnOOQ iC 22 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS ^'^'^^ r Probable error of E^ = -67449 x/N (1 - Ei*/N») .... {\xvt.)\ E^ = -67449 n/N(1 - E^/N*) .... (Ixv.). ^ Correlation between errors in E. and E, = - a/ J! " "^^(S! i! T & S-i • • (l^vL). ^ * V (1 + i-i/N) (I + Ej/N) ^ ' , Tj , , , . -67449 sin D*/D(^:^D) • /, •• x Probable error m r = ^ r .... (Irvii.), where D = ' * ^ {cf. Sheppard, loc. cit., p. 148). Probable error in ratio Ci/o-j = •67449 o-^TTf/, E,«\. oi%A L fn E,2\. 2./E,^\ The application of the method here discussed to statistics without quantitative scale can now be indicated. If the characters we are dealing with have the same scale, although it be unknown, then, if the quantitative order be maintained, i.e., individuals arranged in order of lightness or darkness of coat or eye-colour, the diagonal line on the table at 45° will remain unchanged, however we may suppose parts of the scale to be distorted, for the distortion will be the same at corresponding points of both axes. Further, if we suppose the mean of the two characters to be the same, this 45° line will pass through that mean, and will serve for the line AB of the above investigation. In this case we must take tan ^ = 1, and consequently (Ixi.) becomes iTj(r.2 = cos y f )/cos (^fj (Ixix.). We can even, when the mean is a considerable way off the 45° line, get, in some cases, good results. Thus, the correlation in stature of husband and wife worked out by the ordinary product moment process is '2872. But in this case E^ = 382*062 Ej = 806'425, and this gives the correlation '2994. On the other hand, the actual ratio of variabilities is 1*12, while (Ixix.) makes it 276 ! This arises from the fact that the errors in E^ and E2, due to the mean being off the 45° line, tend to cancel in El + E^, but tend in directly opposite directions in the ratio of the cosines. Similarly the correlation between father and son works out *5666, which may be compared with the values given in Illustration V. below, ranging from '5198 to '5939. Again, correlation in eye-colour between husband and wife came out by the excess process •0986, and by the process given earlier in the present Memoir '1002. But all these are favourable examples, and many others gave much worse results. We ought really only to apply it to find cja-^ when the means are on the 45° line, as in the correlation of the Google Digitized by -1 TO THE THEORY OF EVOLUTION. 23 same character in brethren, and even in this case the statistics ought to be collected ad hoc, i.e., we ought to make a very full quantitative order, and then notice for each individual case the number above and below the type. For example, suppose we had a diagram of some twenty-five to thirty eye tints in order {e.g., like Bertrand's), then we take any individual, note his tint, and observe how many relatives of a particular class — ^brethren or cousins, say — have lighter and how many darker eyes ; the difference of the two would be the excess for this individual. The same plan would be possible with horses' coat-colour and other characters. After trying the plan of the excesses on the data at my disposal for horses' coat-colour and human eye- colour (which were not collected ad hoc), I abandoned it for the earlier method of this Memoir ; for, the classification being in large groups, the proportioning of the excess (as well as the differences in the means) introduced too great errors for such investigations. § 7. On a Generalisation of the Fundamental Theorenx of the Present Memoir. If we measure deviations in units of standard deviations, we may take for the equation to the correlation surface for n variables N z = (27r)VE e-^W^^)-H^)} (Ixx.), where K = 1 Til 1 '23. ^n-1, 1 ^,2 '»-i, » rn- n— 1, n and 'Rpg is the minor obtained by striking out the ^th row and qth column, rpg is, of course, the correlation between the pth and ^th variables, and equals r^. S^ denotes a summation for s from 1 to n, and Sg a summation of every possible pair out of the n quantities 1 to n. Now take the logarithmic differential of z with regard to r^ We find For dB,/dr^ = 2Kp^ Digitized by VnOOQ iC 24 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS and, generally, whether 5 is or is not = s, or these are or are not = p and g, we have dir[-R)= n* O^xi.). This follows thus : dr„ \ K / R drp, R2 dvp^ " R 1^ R2 ' or we have to show that djxggt £t]\„i \xpq — Kj»t R^#/ — Rpi* Ry* dvpq "" R R«l^lVg — RyjRyjf , R<^Ryy — fiw/ Ryj ~ R "*" R where jb^R«/ is the minor corresponding to the term r^q in R,^, and q^s$> the minor corresponding to the term r^p.* But this last result is obvious because R^^ only con- tains Vpq in two places, i.e.y as Vp^ and r^p. Putting s =s\ we have the other identity required above, i.e.^ drpqXHJ R9 Returning now to the value for - - — on the previous page, we see that the two z dvpq sum terms may be expressed as a product, or we may put -4=-^+«.(l-)xs.(l-) Now write (2Tr)'VK' z =- .„ ... /:rt e-*. Tx 1 dz d^(h . d(b d(b Hence - -— = — — ^ + ^ -:^ z dVpg dxp dj:^ dxp dxg Now differentiate log z with regard to x^. Then dz d<l> dosp dXp * See also Soott, * Theory of Detenninante,' p. 59, Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 25 Thus finally d,uyixj = — dz d4> djCg do:p 1 z (Pz dz ^4> , dX^g dh d^d^ dxp dxg dr„ dayfajj • (Ixxii.). In other words, the operator djdr^ acting on z can always be replaced by the operator d^jdxpdxq. Let d/dppq denote the effect of applying the operator d/dvp^ to «, and putting Vpg zero after all differentiations have been performed, then the effect of this operator will be the same as if we used d^jdxpdxq on 2;, putting Vpq zero before differentiation. Generally, let F be any series of operations like d/dvp^^ then we see that \drpq' dryq, ' ipq Uipig, dVp..^,, \ dXj/d.Cq dxpdxq, ' d^Vp,dXg„ ' / (27r)** (x.») Now let F be the function which gives the operation of expanding z by Maclaurin's theorem in powers of the correlation coefficients, i.e., F = e««^-'5^)> then z = e ^(-' .7^^ z = — M-' 1^) e - »«'<'-^ . This is the generalised form of result (xiv.) reached above. Now let Zr. — ■ ^ • ^-iSi<''"> nov. iGT, Zq — ^2^^^^ € , then z^^ is the ordinate of a frequency surface of the nth order, in which the distribution of the n variables is absolutely independent. We have accordingly the extremely interesting geometrical interpretation that the operator applied to a surface of frequency for n independent variables converts it into a surface of frequency for n dependent variables, the correlation between the sth and sth variables being r„,* ^ I should like to suggest to the pure mathematician the interest which a study of such operators would have, and in particular of the generalised form of projection in hyperspace indicated by them. VOL. CXCV. — A. E Digitized by VnOOQ iC 26 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Expanding, we have + • • • +||{Sa(»'«'d^,)}"^o + . •. . ■ . . (Ixxiii.). Our next stage is to evaluate the operation Hj'*''d^ Co- llet us put ,Vi — x„ ,«8 = a?/ — 1, ,V3 = a;,(a;/ — 3), and ,Vp = the pth function of «, as defined by (xv.). Let €, be a symbol such that c/ represents ,Vp. Then we shall show that ^^"•^.j =^o = «o{s,(r,^^^)}"' (Ixxiv.). We shall prove this by induction. By (xii.) ,v^+i = X, ,Vm — m ,t;«.i, or c/"*"^ = X, c/ — m c/""\ and by (xvi.) ^ = m,i;«_,. or — = m e,- ^ Now, let X (€*) be any function of c. if we suppose it can be expanded in powers of c, Then d = S(A,<?e,»-i) = S(A,(x,€,»-e/+i)) = a!^(A, €,1) - €. S(A, €/) = («*-«') X(«') (Ixxv.). Similarly ;^;;j^ x(«* .«'') = (^'-«')(^'—«'') x(«« «*') • • • • (Ixxvi.). Now suppose that Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 27 then {^«(^-d^)}'""^o = S,(^-^>oU, where U stands for. {S2(r,^€,€^)}*. Hence, remembering that dzjdx^ = — z^t>y = «oS»(»'«.<»e^)U' = 2o{S2(r^€^^)}-+', which had to be proved. But it is easy to show by simple differentiation that ^ Zo {^i{r^^,)Y (Ixxvii.). Hence the theorem is generally true. Thus we conclude that + • • • + |4 {S2(^'^'*.)}'"+ . • • •] O^^viii.). It is quite straightforward, if laborious, to write down the expansion for any number of variables. Now let Q be the total frequency of complices of variables with x^ lying between h^ and 00 , X2 between h^ and oo , . . . aj, between h, and oo , . . . 5c„ between h^ and oo ; and let Qq be the frequency of such complices if there were no correlations. Then I ... I ... 2; dx^ dx^ . . . dxt dx^ fCD •flO ttOO ftO I ... I ... 1 ZQdx^dxci . . . dx, . . . dXf^ hi JA, Ja. Ja. Now let £ 2 Digitized by VnOOQ iC 28 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS where We have Qo = 'iHfiA . . A • • i8« H^ Hg . . H, . . H,. But by (xviii.) where and as above, Thus ,W/,_l = [,Uj,_l]^ = ». (Ixxx.), ^, = ^'e-i^dx./e-i'"\ (Ixxxi.). (vfel" 1 • • • 1 • ' * L '^^'' V'-'V- • • e-*<"'+'^+ . . . +x^+ . . . ^-.^dxidx^. ..dx,...dx„ = H,H, . . . H, . . . H^A . . . i8, . . . /8/%i4=i4^ . . . , Ps Pr Pt" or j j . . . j . . . j ZqU{,v^) dx^dxc^ . . . dx, . . . dxn = QqH r^) . . . . (Ixxxii.). where 11 denotes a product of ,Vj, for any number of v's with any s and p. The rule, therefore, is very simple. We must expand the value of z ini/& as given by (Ixxviii.) above, then the multiple integral of this will be obtained by lowering every v's right- hand subscript by unity (remembering that sVq =1), and further dividing by the fi of the left-hand subscript. The general expression up to terms of the fourth order has been written down ; it involves thirty-four siuns, each represented by a type term All these would only occur in the case of the correlation of eight organs, or when we have to deal with twenty-eight coefficients of correlation. Such a number seems beyond our present power of arithmetical manipulation, so that T have not printed the general expressions. At the same time, the theory of multiple correlation is of such great importance for problems of evolution, in which over and over again we have three or four correlated characters to deal with,'*^ that it seems desirable to place on record the expansion for these cases. I give four variables up to the fourth and three variables up to the fifth order terms. Afterwards I will consider special cases. * In my memoir on Prehistoric Stature I have dealt with five correlated organs, *.«., ten coefficients. In some barometric investigations now in hand we propose to deal with at least fifteen coefficients, while Mr. Bramley-Moore, in the correlation of parts of the skeleton, has, in a memoir not yet published, dealt with between forty and fifty cases of four variables or six coefficients. Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 29 Vcdue of the Quadruple Integral in the Case of Four Variables.^ Qo fil^% A^3 ^1^4 ^3^3 ^2^4 ^8^4 ^1^3^8 ^ ^1^2^3 ^ A^2^3^4 A^2^3^4 ^^18^*84 //' I ^!li^J4 ,, iv _j_ ^^'28^'8. ^ ^^18^^84 ^ /// _^ fhj^u t; >^ + fVM <y ' + ^^*24^84 «i iv I ^^12^*24 «. " ^_ ^^18^*24 ^2^3^4 ^ AA^4 ^ ^i^2^3^4 i 2^14^24 iv I ^^28^24 //I +/3i/32/34 ''^ + fiJ3^, ""' J + 1^1/31/8/^''^ ^ Ms'"'''' +/8i/8/^^^ +/8a''^'^ +/92/8/^^^ + /3A '* ''^ + M^ '''''' + Ay8A ''^''^ + /^^ ''^''^ ^^l^^2^>3 P1P2P8 P1P2P3 3^13?^ ,„ „ 3r^/ro3 , .^, ^^'u^'23^ " /// + A/8^A ''^ "^ + /9i/9^. ''^ "' + A/3A "^ ''' * To simplify the notation, »,', »,", «;,'" »,'» have been used for i»„ jf,, 8»«, 4»»- Digitized by VjOOQ IC 30 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS + /8A/9A ^ ^ +/3i/S,/93^4 ^ > +/8i/8^A ^^ 6rigrggr34 , ,„ 6r)3r33^^^ ,„ 6ri4rg8r^ ,„ .^ + )8,^A ^ ^ ^ + /3i;SA ''^''^ "^^ J + liW, " * +i8ii9s '''» +A/8/'' ' +/3W38 * ' +.8^4'*^ ^/Sa'^''^^ ^ i8i/8^8 "^^ '■' ^ iSi^A "^^ ' + ^A^4 "^^ ' ^ /e./8A/84 ''^ ''« + /Sii8,i94 * '^ + i8>^A ' '^ + /8i/3w38^« "^'"^^ + fi^M^. '' ^' + /3i/9A ^^ + i8i/8A ^ '^ ^ /3^A " * ^ /9,/88i8. " * , ^^'l8 ^34 / // , ^ 4?j£s/_ „, j^ ^^'l8^ ^SA /// / i ^^'l8^84^ /// jy , ^14' ^84 / iy , ^^uV „ '// w , 4rj/»^ „ „ 4rg3»^ ,„ j^ + /Si/Ss-S, ^«^« + A/8A ' ' + /8^A ' ' ^ M^* ' ' + ^AA "' "' + ^. "' "• + ^ "• ' . 6r„ Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 31 I -^--^^'la ^14^*24 f ft \v % -*■ ^^12^14 ^'24 / // iv I "'••^^12^14^91 / // ;» + A)8A ^'^^^ ^» + A/8A ^ ' -^ ^ A/S3/S, ^'»^* ^« IV + A/SW3A ^ » * + /3i/8^A ' ' + /3i/3^A ^ ' ^ » Digitized by VjOOQ IC 32 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS ^ /Ss/8A '''' ""^ ""' + /8^,/3, *^ ""' ''« + /9^,/3, ''l ''^ ''^ , 2^12 ^18^14^23 / // m , 247-13 ^13^1^ , „ ;^ 24rigri3rayi4 / ^/ //' ^ /8i/32)8A "^^"^^"^^ + /8,/82/33/8, ^'^^^ ^^ ^- ^1/8,^3/3, ^^ "« '^^ + i9ii82/9A ^ '^ ^ + /9i)82/9A ^1^1^^ ^1 + /81/3,^3/S, ^«^i ^1 + /3i;8,^3/3, ^^^i"« + /9i/33)S3/3, ^^^^^1 ^1 + /3i;8,/33/9, ^'^ ^* ^^ + A/S^a^* ""^'^ ""' + Ay92/9A ^^^1^^ + fi^^^, ^1^1 ^'^ . . . . (Ixxxiii.). In the case of three variables, we must cancel in the above expression all terms involving jS.^ Thus we shall have 3 instead of 6 first order terms, 6 instead of 21 second order terms, 10 instead of 56 third order terms, and 15 instead of 126 fourth order terms — a much more manageable series. I give below the extra term necessary for calculating the value of (Q — Qo)/Qo a.s far as the fifth order terms in the case of three variables. Fifth Order Terms for Three Vanables. PiPiP% PiPA PiPoPa ins** iHtS*"^ lO'S'*'* + 20r. f/t + ^^A'"^-^^''^« + /3./3A '^^^''^ I ^^''''''^'^•^• Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 33 A numerical illustration of these formulsB will be given in the latter part of this Memoir. It will, however, be clear that what we want are tables of log ( ^ j, including log ('^) or log [—1 for a series of values of h. Such tables would render the compu- tation of — jr-^ fairly direct and rapid ; they could be fairly easily calculated from existing tables for the ordinate and area of the normal curve, and I hope later to find some one willing to undertake them. Meanwhile let us look at special cases. In the first place, suppose, in the case of three variables, that the division of the groups is taken at the mean, ?.e., ^^ = A^ = A3 = 0. Then we have /8i = A = )83 = fV*''<^=»=A/|- v/ = v{ = v/" = < = < = <'= -1 < = v^' = <' = < = < = <" = 3. Hence we have = Qojl + — (sin-irij + sin-'ris + sin-^rja)! (Ixxxv.). Let ri2 = cos D^, rjg = cos D^g, r^ = cos Djg, and let E be the spherical excess of the spherical triangle whose angles are the divergences D^, Djj, D23. Then we have Q — Qo ^ _ ?E T\ r» r\ —.'^ Qo 2 - 2 ~ ^i» ~ ^^3 ~ ^«8 - 2 Or: sin^-^J=cosE (Ixxxvi.). Now take the case of four variables. Here we have ^1 = ^2 = ^3 = ^4 = V 5 < = V = <" = 1Jji» = 1 and all the odd v's zero. Hence VOL. cxcv. — ^A. p Digitized by VjOOQ IC 9^?3^S4 34 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Q 2' / 2 \® — Q— ^ = ;. ('-12 + ''js + »*it + ^23 + »'24 + r^) + [-) ('•w'as + rijra, + r^^r^^) *> 1 / 2 \® + ^ ]3 (**•«' + ''is" + '■"' + ''2S' + ^2*' + **»*') + (^ j (*'18'*13'*14 + ^12* /2 V 1 + '-Isna'M + »*]*''24»*84) + \^y i3"(^l*'*'28 + '•l4»'23' + ^3^24* + ^3% + ^-llV,^ / 2 \^ 1 + ^12^3*^) + [yj 12" (^12^^14^23 + ^Au'^SS + ^12^''l3^54 + ^18^'u^^24 + ^13^23^^24 + ^4^23^ + ^2^13^4 + ^12^^34 + ^2^23^^*34 + ^14^28^34^ + ^2^24^^^ + ^13^24^34^) + (IxXXvii.). This is the correct value including terms of the fourth order, but to this order of approximation we can throw it into a much simpler form. Let r,y = sin 8,^, then — p-— ^ ^ = sin"^ rjg + sin"^ rjg + sin~^ r^^ + sin"^ Vc^ + sin"^ r^^ + sin~^ r^ 2 . + — (sin"^ 7*13 sin""^ r^^ sin""^ r^^ + sin"^ r^^ sin"^ 7\2Z ^^^"^ ^24 + sin~^ r^g sin~^ r^^ sin""^ r^ + sin""^ r^^ sin"^ r^^ sin"^ r^^) + ~ [sin-» n* sin-^ r«,{(l - r,,«) (1 - n,') (1 - r^») (1 - r,,*)]-* + sin-i ri^sin-i r3,{(l - r,,^) (1 - r,,*) (1 - r^') (1 - V)}"* + sin-i r,3 sin-i r,,{(l - V) (1 - r,,«) (1 - V) (1 - rg/)}"*] = 8j3 + 8i3 + 8j4 + 823 + 824 + ^34 + ^ (812813^14 + 813833824 + 813823834 + 814824834) 2^ A^ggg COS 8|^ COS 8g3 4- 8128^ COB 812 COS 83^ 4- SigSg^cofr 8,^00382 A ^ \ COS 813 COS Si3 cos Si4 COS Sjj cos 834 COS Sjj4 / . . . • (Ixxxviii.). We may write this sin — ^— ^ ^ = cosE' (Ixxxix.) where E' = "2 "■ 812 — 813 — 814 — 823 — 824 — 834 — ^ (812833814 + 810823834 + 813823824 + 814824834) 2 / 8i^833 cos 834 cos 823 -f 8,38^005812008834 4- 813824 cos 8i3 COB 83t \ IT \ COB 8j3 COS Si3 COS 834 COS 853 COS 834 COS 834 / The expressions E and E' of (Ixxxvi.) and (Ixxxix.) are of considerable interest, for they enable us to express the area of a spherical triangle in three-dimensioned space, Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 35 and (up to the above degree of approximation) the volume of a " tetrahedron " on a " sphere " in hyperspace of four dimensions. In fact, the whole theory of hyperspace " spherical trigonometry " needs investigation in relation to the properties of multiple correlation. In our illustrations (viii.) and (ix.) will be found examples of the above formulae applied to important cases in triple and quadruple correlation in the theory of heredity. I consider that the formulae above given will cover numerous novel applications, for many of which greater simplicity will be introduced owing to the choice of special values for the h's or for the correlation coefficients. (8.) Illustrations of the New Methods. Illustration I. Inheritance of Coat- colour in Horses. — The following represents the distribution of sires and fillies in 1050 cases of thoroughbred racehorses, the grouping being made into aU coat-colour classed as ** bay and darker," " chesnut and lighter":— Colour. Sires. Bay and darker. Chesnut and lighter. FiUiea. Bay and darker . . . 631 125 756 Chesnut and lighter . 147 147 294 778 272 1050 a h a + b c d c + d a + c b + d N Then we require the correlation between sire and filly in the matter of coat-colour, and also the probable error of its determination. We have from (iv.) and (v.) «i = ^ \^ -^ = \/- \e-^^dx = •481.905, «2 = ^ V^ ^ = V ^Jo^ '^^y = -440,000. Hence from the probability integral tables h = -64630, k = -58284. We have then : log HK = T-037,3514 by (xvii.), F 2 Digitized by VjOOQ IC 36 PROFESSOE K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Thence c = "'~J = -619,068 from (xxi.). Calculating out the coefficients of the series in r in (xix.) we find •619,068 = r + •188,345r2 + •064,0814r« + •107,8220r* + '005,99867^ + •067,2682r« 4- &c. Neglecting powers of r above the second, we find by solving the quadratic and taking the positive root r — -5600. Solving by two approximations the sextic we finally determine . r = -5422, correct, I think, to four places of figures. Turning now to the probable error as given by Equation (1.), I find ^s + jfc2 _ 2rhk = -348,924, and from (xlix.) logxo= 1*170,0947. Further : '^ll\^ ~ -275,642 , Jpi^ = '393,078. Hence from (xlvii.) and (xlviii.) we find 1 r-»93,078 1 r -275,642 and by means of the probability integral table i|ri = -108,884, xji^ = -152,865. By substituting in (1.), we find probable error of r = '0288. From (xxxiv.) and (xxxv.) we find p.e. of A = -0282. p.e. oi k = -0278. Thus, finally, we may sum up our results h = -6463 ± -0282, k = '5828 ± '0278, r = -5422 ± -0288. The probable en-or of this r, if we had been able to find it from the product moment, would have been '0147, or only about one-half its present value. Digitized by VnOOQ iC TO THE THEORY OF EVOLUTION. 37 Illustration IL — Our analysis opens a large field suggested by the following problem : — What is the chance that an exceptional man is born of an exceptional father f Of course much depends on how we define " exceptional," and any numerical measure of it must be quite arbitrary. As an illustration, let us take a man who possesses a character only possessed by one man in twenty as exceptional. For example, only one man in twenty is more than 6 feet 1'2 inches in height, and such a stature may be considered " exceptional." In a class of twenty students we generally find one of " exceptional " ability, and so on. Accordingly we have classed fathers and sons who possess characters only possessed by one man in twenty as exceptional. We first determine h and Jc, so that the tail of the frequency curve cut off is -^ of its whole area. This gives na h = k = 1*64485. Next we determine HK = ^e'^'''^'*\ and find log HK = 2-026,8228. Then we calculate the coefficients of the various powers of 7* in (xix.). We find \og^hk= -131,2225. logi(/t« - 1) (F - 1) = 1-685,5683. log^Ah^ - 3)(^ - 3) = 3-990,1176. logxio (A* - 6/i* + 3)(** - 6F + 3) = 1-464,4772. log^(A* - 10A« + 15)(** - lOP + 15) = 2-9.25,6367. It remains to determine what value we shall give to r, the paternal correlation. It ranges from "3 to "5 for my own measurements as we turn from blended to exclusive inheritance. Taking these two extreme values we find — ^r=r- = -0046344 or '0096779. But — 2 = — — ^ :^ -^ and the second term is the chance of exceptional fathers with exceptional sons, when variation is independent, i*e., when there is no heredity, = ^ X Vo = '0025. Thus d/N = -007134 or -012178 ; accordingly 6/N = -042866 or -037822. Hence we conclude that of the 5 per cent, of exceptional men -71 per cent, in the first case, and 1*22 per cent, in the second case, are bom of exceptional fathers, and 4-29 per cent, in the first case and 3-78 per cent, in the second case of non-exceptional lathers. In other words, out of 1000 men of mark we may expect 142 in the first case. Digitized by VnOOQ iC 38 PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS 244 in the second, to be born of exceptional parents, while 858 in the first and 756 in the second are born of undistinguished fathers. In the former case the odds are about 6 to 1, in the latter 3 to 1 against a distinguished son having a distinguished father. This result confirms what I have elsewhere stated, that we trust to the great mass of our population for the bulk of our distinguished men. On the other hand it does not invalidate what I have written on the importance of creating good stock, for a good stock means a bias largely above that due to an exceptional father alone. In addition to this the •^- of the population forming the exceptional fathers pro- duce 142 or 244 exceptional sons to compare with the 858 or 756 exceptional sons produced by the ^| of the population who are non-exceptional. That is to say that the relative production is as 142 to 45*2, or 244 to 39*8, i.e., in the one case as more than 3 to 1, in the other case as more than 6 to 1. In other wordsy exceptional fathers produce exceptional sons at a rate 3 to 6 times as great as non-exceptional fathers. It is only because exceptional fathers are themselves so rare that we must trust for the bulk of our distinguished men to the non-exceptional class. Illustration III. Heredity in Coat-colour of Hounds. — To find the correlation in coat-colour between Basset hounds which are half-brethren, say, ofispring of the same dam. Here the classification is simply into lemon and white ijiiv) and lemon, black and white or tricolour (f), The following is the table for 4172 cases : — Colour. t. IV). Totals. t. 1766 842 2608 Iw. 842 722 1564 Totals 2608 1564 4172 Proceeding precisely in the same way as in the first illustration we find «! = Oj = h=zh = 25024 318,957 157,6378 logKH=:l- c= -226,234. It will be suflficient now to go to r*. We have •226,234 = r + '050,867 r* + -134,480 r» + '035,587 v\ Digitized by VjOOQ IC TO THE THEORY OF EVOLUTION. 39 The quadratic gives r = '2237. Using the Newtonian method of approximating to the root we find r = -2222. Summing up as before, after finding the probable errors, we have h = k= -3190 ± -0133, r= -2222 ±-0162. Illustration IV, Inheritance of Eye-colotir in Man.— To find the correlation in eye-colour between a maternal grandmother and her granddaughter. Here the classification is into eyes described as grey or lighter, and eyes described as dark grey or darker.* Tint. Maternal grandmother. Totals. Grey or lighter. Dark grey or darker. 1 1 1 C5 Grey or lighter .... 254 136 390 Dark grey or darker . , 156 193 349 Totals 410 329 739 As before, we find aj = -109,607, ajj = -055,480, h = -138,105, k = -069,593, log HK = T-196,6267, c = -323,760. Series for r up to r* •323,760 = r + -004,806?^ + -162-696r5 + '000,358^* The quadratic gives r = "3233, and the biquadratic r = -3180, the value of the term in r* being -000,00366, so that higher terms may be neglected. Determining the probable errors as in Illustration I., we sum up : — * According to Mr. Galton's classification, the first group contains eyes described as light blue, blue, dark blue, blue-green, grey ; and the second eyes described as dark grey, hazel, light brown, brown, dark brown, verv dark brown, black. Digitized by VnOOQ iC 40 PKOFESSOE K. PEAESON ON MATHEMATICAL CONTBIBUTIONS h = -1381 ± 0312, k = -0696 db 'OSll, r = -3180 ± -0361. Illustration V. Inheritance of Stature. — The following data have been found for the inheritance of stature between father and son from my Family Data cards, 1078 cases : — Mean stature of father. . . . 67"-698 son 68''-661 Standard deviation of father . . 2"7048 son. . . 2"7321 Correlation = -5198 ± '0150. Now for purposes of comparison of methods the correlation has been determined for this material from various groupings of fathers and sons : — (A.) Fathers. « Class. Below 67"-5. Above 67"-5. Totals. Below 67"-6 . . 269-25 96-75 365 Above 67"-5. . 232-25 480-75 713 Totak . . . 501-6 576-5 1078 (B.) Fathers, CQ Class. Below 66"-5. Above 66"-5. Totals. Below 67"-5 . . 211-25 153-75 365 Above 67"-5 . . 152-75 560-25 713 Totals . . . 364 714 1078 Digitized by VjOOQ IC (C.) TO THE THEORY OF EVOLUTION. Fathers, 41 Class. Below 67"-5. Above 67"-5. Totals. Below 68-6" . . 356-26 182-25 638-5 Above 68'5" . . 146-25 394-25 539-6 Totals . . . 501-6 676-6 1078 (D.) Fathers. CQ CSass. Below 68"-6. Above 68"-5. Totals. Below 69"-6 , . 506 182 688 Above 69"-5 . . 149-5 240-6 390 Totals . . . 655-6 422-6 1078 (E.) Fathers. Class. Below 69"-5. Above 69"-6. Totals. Below 70"-5 . . 669 147 816 Above 70"-6 . . 128 134 262 Totals . . . 797 281 1078 (F.) « Fathers, Class. Below 70"-5. Above 70"-5. Totals. Below 69"-5 . . 641-25 46-76 688 Above 69"-5 . . 271-75 118-25 390 Totals . . . 913 165 1078 VOL. OXCV. — A. Digitized by VjOOQ IC 42 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS Table of Results. Claasificatiom. Correlation. Mean of sons. Mean of fathers. A B D E F •5939 ± -0247 ■5557 ± -0261 •5529 ± -0247 -5264 ± -0264 •5213 ± 0294 -5524 ± -0307 k. 68"-64(- -416,32) 68"-64(- -416,32) 68"-50(- -001,16) 68"-53 (-353,71) 68"-60 (-696,57) 68"-63 (-353,71) A. 67"-74(- -087,00) 67"-63(- -418,86) 67"-74(- -087,30) 67"-77 (-274,30) 67"-76 (-641,30) 67"-73 (1-023,44) Now these results are of quite peculiar interest They show us : — (i.) That the probable error of r, as found by the present method, increases with h and k But the increase is not very rapid, so that the probable errors of the series range only between '025 and •031. Hence while it is an advantage, it is not a very great advantage, to take the divisions of the groups near the medians. It is an advantage which may be easily counterbalanced by some practical gain in the method of observation when the division is not close to the medians. (ii.) While the probable error, as found from the present method of calculation, is 1*5 to 2 times the probable error as found from the product moment, it is by no means so large as to seriously weigh against the new process, if the old is un- available. It is quite true that the results given by the present process for six arbitrary divisions diflPer very considerably among themselves. But a consideration , of the probable errors shows that the differences are sensibly larger than the prob- ; able error of the differences, even in some case double ; hence it is not the method , but the assumption of normal correlation for such distributions which is at fault. As ' we shall hardly get a better variable than stature to hypothesise normality for, we see the weakness of the position which assumes without qualification the generality of the Gaussian law of frequency. (iii.) We cannot assert that the smaller the probable error the more nearly will the correlation, as given by the present process, agree with its value as found by the product moment. If we did we should discard '5213, a very accordant result, in favour of '5529, or even '5939. The fact is that the higher the correlation the lower, ceteris "paribus^ the probable error, and this fact may obscure the really best result. Judging by the smallness of h and k and of the probable error, we should be inclined to select C or the value '5529. This only differs from '5198 by slightly more than the probable error of the difference ('033 as compared with '029) ; but since both are found from the same statistics, and not from different samplings ot the same population, this forms sufficient evidence in itself of want of normality. The approximate character of all results based on the theory of normal frequency must be carefully borne in mind ; and all we ought to conclude from the present Digitized by Google TO THE THEORY OF EVOLUTION. 43 data for inheritance of stature from father to son would be that the correlation = '55 ± '015, while the product moment method would tell us more definitely that its value was '52 ± -015. There is no question that the latter method is the better, but this does not hinder the new method from being extremely serviceable; for many cases it is the only one available. Illustration VI. Effectiveness of Vaccination. — To find the correlation between strength to resist small-pox and the degree of eflfective vaccination. We have in the earlier illustrations chosen cases in which in all probability a scale of character might possibly, if with difficulty, be determined. In the present case, the relationship is a very important one, but a quantitative scale is hardly discover- able. Nevertheless, it is of great interest to consider what results flow from the application of our method. We may consider our two characters as strength to resist the ravages of small-pox and as degree of efiective vaccination. No quantitative scales are here available ; all the statistics provide are the number of recoveries and deaths from small-pox, and the absence or presence of a definite vaccination cicatrix. Taking the Metropolitan Asylums Board statistics for the epidemic of 1893, we have the table given below, where the cases of " no evidence " have been omitted. Proceeding in the usual manner we find a^= -86929 a^ = -54157 h= 1-51139 *= -74145 c = -782454. Hence the equation for r is •782,454 = r + '560,310?'^ - -096,3787^ + -081,8817^ - •000,172?'^ - •040,0597'^^ whence r = '5954. Simiming up we have, after calculating the probable errors, h= 1-5114 ± -0287, k = -7414 ± -0205, r = -5954 db -0272. Strength to resist Small-pox when incurred. g g Cicatrix. Becoveries. Deaths. Total. Present 1562 42 1604 Absent 383 94 477 Total 1945 136 2081 O 2 Digitized by Google 44 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS We see accordingly that there is quite a large correlation between recovery and the presence of the cicatrix. The two things are about as closely related as a child to its " mid-parent." While the correlation is very substantial and indicates the protective character of vaccination, even after small-pox is incurred, it is, perhaps, smaller than some over- ardent supporters of vaccination would have led us to believe. Illustration VIL Effectiveness of Antitoxin Treatment. — ^To measure quanti- tatively the effect of antitoxin in diphtheria cases. In like manner we may find the correlation between recovery and the administration of antitoxin in diphtheria cases. The statistics here are, however, somewhat diflScult to obtain in a form suited to our purpose. The treatment by antitoxin began in the Metropolitan Asylums Board hospitals in 1895, but the serum was then administered only in those cases which gave rise to anxiety. Hence we cannot correlate recovery and death with the cases treated or not treated in that year, for those who were likely to recover were not dosed. In the year 1896 the majority of the cases were, on the contrary, treated with antitoxin, and those not treated were the slight cases of very small risk ; hence, again, we are in great difficulties in drawing up a table.* Further, if we compare an antitoxin year with a non-antitoxin year, we ought to compare the cases treated with antitoxin in the former year with those which would probably have been treated with it in the latter year. Lastly, the dosage, nature of cases treated, and time of treatment have been modified by the experience gained, so that it seems impossible to club a number of years together, and so obtain a satisfactorily wide range of statistics. In 1897, practically aU the laryngeal cases were treated with antitoxin. Hence the best we can do is to compare the laryngeal cases in two years, one before and one after the introduction of antitoxin. The numbers available are thus rather few, but will help us to form some idea of the correlation. I take the following data from p. 8 of the Metropolitan Asylums Board * Report upon the Use of Antitoxic Serum for 1896 ' : — Laryngeal cases. Eecoveries. Deaths. Totals. With antitoxin, 1896 .... 319 143 462 Without antitoxin, 1894 . . . 177 289 466 Totals 496 432 928 * When a new drug or process is introduced the medical profession are naturally anxious to give every patient the possible benefit of it, and patients of coiu*se rush to those who first adopt it. But if the real efficiency of the process or drug is to be measured this is very undesirable. No definite data by which to measure the effectiveness of the novelty are thus available. Digitized by Google TO THE THEORY OF EVOLUTION. 45 Here I find r = '4708 ± '0292. A further table is of interest : — Laryngeal cases. Kequiring tracheotomy. Not requiring it. Totals. Without antitoxin, 1894 . . . 261 205 466 With antitoxin, 1896 .... 188 274 462 Totals 449 479 928 In this case we have r = "2385 ± -0335. Lastly, I have drawn up a third table :— Total Infantile Cases, Ages — 5 years. Recovery. Death. Totals. With antitoxin, 1896 . . . 912 434 1346 Without antitoirin, 1894 . . 615 556 1171 Totals 1527 990 2517 Here we have* r = '2451 ± 0205. The three coeflficients are all sensible as compared with their probable errors, and that between the administration of antitoxin and recovery in laryngeal cases is substantial. But the relationship is by no means so great as in the case of vaccina- tion, and if its magnitude justifies the use of antitoxin, even when balanced against other ills which may follow in its train, it does not justify the sweeping statements of its eflfectiveness which I have heard made by medical ifriends. It seems until wider statistics are forthcoming a case for cautiously feeling the way forward rather than for hasty generalisations. Illustration VII f. Effect on Produce of Superior Stock. — To find the eflfect of superiority of stock on percentage goodness of produce. To illustrate this and also the formula (Ixxxiii.) for six correlation coefficients, we wiU investigate the eflfect of selecting sire, dam, and one grandsire on the produce when there * The values of r for all the three cases of this Illustration were determined with great ease from Equation (xxiv.). Digitized by Google 46 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS is selective pairing of dam and sire. We will suppose grandsire, dam, and sire to be above the average, and investigate what propoi-tion of the produce will be above the average. As numbers very like those actually occurring in the case of dogs, horses, and even men, we may take Correlation of grandsire and offspring . = '25 „ sire or dam and offspring = "5 in both cases „ sire and grandsire . . . = "5 Selective mating for sire and dam. . . = "2 We will suppose zero correlation between paternal grandsire and dam, although with selective mating this may actually exist.* We have then the following system : — ^14? ^^ 25, r^ = '5, r^^ = '5, t^ = '2, Vi^ := '5, r^^ = 0. Hence, substituting these values in (Ixxxvii.), we find — ^after some arithmetic : (Q-Qo)/Qo= 1-4851. But Qo is the chance of produce above the average if there were no heredity between grandsire, sire, and dam, and no assortative mating. N Hence it equals iX^X^X^N^— .-. Q= '1553 N. Or, of the produce '5 N above the average, '1553 N instead of '0625 N are bom of the superior stock owing to inheritance, &c. In other words, out of the '5 N above the average, '1553 N are produced by the stock in sire, dam, and grandsire above the average, or by '1827 of the total stock, t The remaining '8173 only produce '3447 N, or the superior stock produces produce above the average at over twice the rate of the inferior stock. Absolutely, the inferior stock being seven times as numerous produces about seven-tenths of the superior offspring. Illustration IX, Effect of Exceptional Parentage. — Chance of an exceptional man being bom of exceptional parents. Let us enlarge the example in Illustration II., and seek the proportion of exceptional men, defined as one in twenty, born of exceptional parents in a community with assortative mating. * A correlation, if there be substantial selective mating, may exist between a man and his mother-in- law. Its rumoured absence, if established scientifically, would not, however, prove the non-existence of selective mating, for A may be correlated with B and C, but these not correlated with each other. t The proportion of pairs of parents associated with a grandsire above the average was found by putting -5, % and for the three correlation coefficients in (Ixxxv.). In comparing with Illustration II., the reader must remember we there deiilt with an exceptional father, 1 in 20, here only vdth relatives above the average — a very less stringent selectio^Fi. Digitized by Google TO THE THEOEY OF EVOLUTION. 47 Here we taJce for father and son r^^ = "5, for mother and son i\^ = '5, and for assortative mating, r^^ = '2. We have then to apply the general formulae (Ixxxiii.) and (Ixxxiv.) for the case of three variables. We have Ai =^2 =^3 = 1-64485 A = ^9 = iSa = -484,795 Ui' = V = V" = 1 '644,850 Vj' = V," =: t>g"' = 1-705,532 v,' = <' = V = — -484,356 v^' = <' = V' = - 5-913,290 Whence, after some arithmetical reduction, we find (Q - QoVQo = 20-0389. ButQo = ^X^X-^N = TT^tf^ N. Hence Q = -00263 N. We m\ist now distinguish between the absolute and jllative production of excep- tional men by exceptional and non-exceptional parents. The exceptional pairs of parents are obtained by (xix.), whence we deduce, putting r = -2, ^ = i = 1*64485, '^-^ _ A (d + b)(d + c)_ d 1 _ .o^oy, . Whence the number of pairs of parents, both exceptional = -005245 N. Thus, '005245 N pairs of exceptional parents produce '00263 N exceptional sons, and '994755 N pairs of parents, non-exceptional in character, produce '04737 N exceptional sons, i.e., the remainder of the -^ N. The rates of production are thus as '5014 to '0476. Or : Pairs of exceptional parents produce exceptional sons at a rate more than ten times as great as pairs of non-exceptional parents. At the same time, eighteen times as many exceptional sons are bom to non-exceptional as to exceptional parents, for the latter form only about J per cent, of the community. The reader who will carefully investigate Illustrations II., VIII., and IX. will grasp fully why so many famous men are born of undistinguished parents, but will, at the same time, realise the overwhelming advantage of coming of a good stock. Digitized by VnOOQ iC Digitized by VjOOQ IC [ 49 ] II. Electncal Conductivity in Gases Traversed by CatJwde Rays. By J. C. McLennan, Demonstrator in Physics^ University of Toronto. Communicated by Professor 5. J. Thomson, F.R.S. Received December 7, 1899— Read February 1, 1900. Though it has been known that a gas becomes a conductor when traversed by cathode rays, yet the laws connecting this electrical conductivity have not hitherto been studied. The theory has been put forward by J. J. Thomson and Rutherford* that when a gas becomes a conductor under a radiation, it does so in virtue of the production of positive and negative ions throughout its mass. This view has been established by their experiments on Rontgenised gases, and confirmed by those of ZELENYf on the same subject. The recent work of Rutherford on Uranium Radiation^ also affords another example of such a process in the gases traversed. The object of the experiments which are described in this paper was to investigate the nature of the conductivity in different gases when cathode rays of definite strength passed through them, and to measure the number of ions produced. With this in view, I have worked with cathode rays produced, after the method of Lenard, outside the discharge tube, as these were found to be more easily dealt with than those inside. The investigation is described under the following subdivisions : — 1. Form of tube adopted for the production of cathode rays. 2. Ionization by cathode rays. 3. Discharging action of cathode rays. 4. Ionization not due to Rontgen rays. 5. Discussion of methods for measuring the ionizations produced in different (es. 6. Description of apparatus used. 7. Explanation of the method adopted for comparing ionizations. 8. Ionization in different gases at the same pressure. 9. Ionization in air at different pressures. ♦ * Phil. Mag.,' November, 1896, p. 393. t *Phil. Mag.,' July, 1898, p. 120. X *Phil. Mag.,' January, 1899, p. 109. VOL. CXCV.— A 263. H 3.11.1900. Digitized by Google 50 MR J. c. Mclennan on electrical conductivity in gases 10. Ionization in a gas independent of its chemical composition. 11. Comparison of ionizations produced by cathode and by Rontgen mys. 12. Summary of results. 1. Form of Tube adopted for the production of Cathode Rays. To produce the rays, a modified form of the tube devised by Lenard,* fig. 1, was used. The disc a which closed the end and carried the aluminium window formed the anode. To hold this disc in position, and to render the joint airtight, recourse was had to sealing-wax, which was allowed to set on the previously warmed glass and metal, after which the parts were made to unite by slightly melting the surfaces and pressing them together. By running round the joint with the pointed flame of a blowpipe, any air bubbles present were removed, and complete union was effected. Joints made in this way were found to hold for any time desired. In making the aluminium window airtight, marine glue could be used, but the ordinary commercial soft wax was found to be more suitable. This was especially so when the experiments were in the tentative state and alterations were fi'equently necessary. The wax melted at a lower temperature than the glue, and besides being much more manageable than the latter, it was also less disagreeable to handle. A coating of it on the sealing wax also prevented cracking. As shown in the figure, the anode was provided with a shoulder round the opening of the window. This was found very convenient when the action of the rays on the J^i^fi air in a partially exhausted receiver such as A was 'oeing examined. The receiver was provided with a similar but larger shoulder, and by slipping it over that on the * * Wied. Ann.,' vol. 51, p. 225 (1894). Digitized by Google TRAVERSED BY CATHODE RAYS. 51 anode and applying a coating of wax, an airtight connection could be readily made without interfering with that which secured the aluminium foil to the disc. This latter connection was effected by placing a thin coating of wax upon the brass disc and gently applying heat after the foil was laid upon it. All the space within the projecting shoulder was then covered with a thick coating of the wax, excepting the central portion of the aluminium. In all the experiments with these tubes the anode was well earthed, as was also the positive terminal of the induction coil used to produce the discharge. As regards the distance between the cathode and the anode, it was found best not to make it too small. Otherwise, the discharge would pass in the tube before the available maximum potential difference was reached. The velocity of the carriers has been shown by J. J. Thomson* to vary with the potential difference between the electrodes, and as a consequence an intense radiation was more readily obtained when the distance between the anode and cathode was considerable. In the case of tubes constructed with a short distance between the electrodes, the device adopted by McCLELLANDt of inserting an air gap in series with the tube very largely increased the intensity of the radiation. The foil used by Lenard for the aluminium window was '003 millim. in thickness. In practice it was exceedingly difficult to obtain such foil free from holes. Aluminium about three times as thick was, however, much better in this regard. The induction coil used in the experiments was, besides, very powerful, and, as a radiation sufficiently intense could be obtained with it, this thickness was used throughout the investigation. 2. Ionization by CatJwde Rays. It has been shown by Lenard| that air, when traversed by cathode rays, acquires the property of discharging electrified conductors against which it may be blown, and that, fiirther, it retains this property for some time after the rays producing it have been cut off. According to the theory of Professor Thomson, the air, when in this state, is ionized, and the discharging action is brought about by a motion of the ions in the gas to the charged conductor. Owing to the separation of the positive and negative ions, recombination can take place but gradually, and this readily explains why the discharging power is retained by the air for some time. In order to show that these positive and negative ions are produced in a gas traversed by the rays, the apparatus shown in fig. 1 was used. The cathode rays issuing from the aluminium window a passed through a narrow tube, 6, into an earth-connected metal chamber, A. B was a disc of brass supported ♦ *Phil. Mag.,' October, 1897, p. 315. t *Proc. Roy. Soc.,' vol. 61, No. 373, p. 227. X * Wied. Ann.,' vol. 63, p. 253 (1897). H 2 Digitized by Google 52 MR. J. c. McLennan on electrical conductivity in gases by an ebonite plug, and surrounded by a guard ring. A wire led from this electrode to one pair of quadrants of an electrometer, and the other pair was put to eai-th. Care was taken to screen off electrostatic induction by surrounding the wire and electrometer with earth-connected conductors. The second electrode, C, also supported by an ebonite plug, was connected by a commutator, D, to one of the terminals of a battery of small storage cells, the other terminal being connected to earth. The tube, 6, was made narrow, and penetrated a short distance into the chamber in oixler to confine the rays to a slender pencil, and to prevent their impinging upon the electrodes. By means of the key, K, the electrode, B, could be put to earth when necessary. With such an apparatus, and no field initially between the electrodes, it was found on exciting the discharge tube and breaking the earth connection, K, that the electrometer gained a small negative charge, which did not go on increasing, but soon attained a limiting value. On the assumption that the cathode rays produce positive and negative ions throughout the gas, the explanation of this is obvious. The cathode rays carried a negative charge into the gas, and set up a field which caused the negative ions to move to the walls of the chamber and to the electrode, B. The charge which the latter soon gained, however, set up a field of its own, and a state of equilibrium was reached when the conduction to the electrode was just equal to that proceeding from it. If, instead of there being no field initially between the electrodes, C was joined to the positive terminal of the batteiy, then the electrode, B, gained a positive charge when the tube was excited, and the rate at which its potential rose depended upon the capacity joined to B and the electrometer. With C joined to the negative terminal of the battery, a similar charging took place, except that in this case the charge accumulated was a negative one. This reversal in the sign of the charge collected may be shown with a field of a few volts a centimetre, and clearly points to the existence of positive and negative ions in the gas. Since the cathode rays themselves carry a negative charge, the presence of these carriers alone in the chamber would account for the negative charge obtained with a negative field. With a positive field, however, these carriers would be attracted to the electrode C, and it seems impossible to explain how the electrode B, under these circumstances, could receive a positive charge unless ions were produced by the rays. 3. Dischargivff Action of Cathode Rays, In connection with the experiments of Lenard,* already referred to, cathode rays were allowed to fall upon a charged conductor surrounded with air at atmospheric pressure. This conductor consisted of a wire attached to a gold-leaf electroscoj^e, ♦ • Wied. Ann.,' vol. 63, p. 253. Digitized by Google TRAVERSED BY CATHODE RAYS. 53 and was placed within a zinc box in which was a small opening covered with a film of aluminium, thin enough to allow the rays to pass through. The end of this wire was placed in front of the window and close to it, with the electroscope clear of the direct path of the rays. The box itself was connected to earth and set in position, with its window opposite that of the discharge tube. Using this apparatus, Lenabd found that positive and negative charges alike were completely dissipated by a single discharge through the tube when the aluminium windows were at any distance up to 4 centims. apart. At greater distances than this a similar but only partial discharging of both kinds of electricity occurred when the same amount of rays was used. This loss of charge was no doubt brought about by means of the ionization in the air surrounding the conductor. The known behaviour of an ionized gas, however, would have led one to expect a somewhat different result, especially in regard to the effect obtained with short distances between the windows. When an insulated metal conductor is placed in air ionized by Rontgen rays, Zeleny* has shown that, owing to the greater velocity with which the negative ions diflRise, this conductor takes up a small negative charge, while the gas itself is left with a positive one. If then the ionizations in the two cases are of the same nature, one would have expected that in Lenard's experiments the wire and electroscope would not, under any circumstances, have been finally discharged completely, but would have been left with at least a small negative charge. When, further, it is remembered that the impinging cathode rays themselves carried a negative charge to the wue, this fact affords an additional reason for expecting such a result. Fi^.H. Now the gold-leaf electroscope, as used by Lenard (Exner's type), was not sensitive to small differences of potential, and it was consequently not a suitable instrument for the detection and measurement of effects of this kind. As the explanation of his results seemed, then, to be connected with this lack of sensibility in the measuring instrument, his experiments were repeated, and a quadrant elec- trometer was used in place of the electroscope. ♦ 'Phil. Mag.,' July, 1S98, p. 13^. Digitized by Google 54 MR J. c. Mclennan on electrical conductivity in gases The arrangement was that shown in fig. 2. A copper wire, terminated by a disc, A, of the same metal, was insulated by ebonite fix)m an earth-connected copper tube B, through which it passed to the electrometer. To this tube there was fastened, as shown in the figure, a large, finely-meshed copper gauze which completely protected the disc from electrostatic induction. The tube B also carried a short concentric cylinder a, made of copper, which could be slid out when desired so as to surround the projecting end of the wire and the disc. On placing this apparatus in front of the aluminium window so that the cathode rays fell on the disc, it was found that, although the rays caused a discharging of positive and of negative electricity, still in no case observed was a negative chai'ge on the disc and wire ever completely dissipated. Negative charges fell, however, to limiting values, represented in some cases by potentials of the order of '25 volt, and then remained stationary. In the case of initial positive charges the discharging was not only complete but the disc also gained this limiting negative charge. A similar charging action was observed when there was no initial charge on the disc. Here the disc was subjected to two influences, namely, the cathode rays canying a negative charge to it and the ionized gas about it acting as a conductor and tending to discharge it. This limiting charge can, then, just as in the case already cited, be looked upon as representing a state of equilibrium in which the convection to the disc was just equal to the conduction away fix)m it. As the electric field produced by a given charge on the disc would vary with the distance between it and neighbouring conductors at a different potential, the conduction from the wire could consequently be increased or decreased according as an earth-connected conductor was brought close to the disc or removed farther from it. If then a means were devised of altering in this way the conduction without altering the intensity of the rays impinging on the disc, the value of this limiting charge could be subjected to definite variations. The sliding cylinder a affoixled a simple means of accomplishing this result. If when the tube was excited a stationary state was reached, with this cylinder shoved well back, and it was then brought forward over the wire and disc, the limiting negative charge at once dropped and assumed a steady but smaller value. In order to restore the charge to its original value it sufficed merely to slide the cylinder back to its former position. Another simple verification of this view was afforded by the use of a blast of air. If when the rays were impinging on the disc a blast of air was directed towards it and at right angles to the rays, the limiting charge at once increased to another limiting valut^, and when the blast stopped it again dropped to its original amount. As the velocity of the cathode rays has been estimated by J. J. Thomson* to be of the order of 10^^ centims. per second, it is clear that any ordinary bliist could produce ♦ * Phil. Mag.,* October, 1897, p. 315. Digitized by Google TRAVERSED BY CATHODE RAYS. 55 very little effect on the motion of these carriers. On the other hand, the velocity of the ions in Rontgenised air has been found by Rutherford* to be about 1 '6 centims. per second, under a field of a volt a centimetre, and consequently of the order of that of the blast. In the experiment described, the effect of the blast, therefore, was to decrease the conduction away from the electrode by removing the ionized gas ; and as no change was made in the intensity of the rays impinging on the disc, this con- sequently produced an increase in the residual charge. This increase, however, did not go on indefinitely, but ceased when the field it set up was sufficient to neutralise the effect of the blast ; hence the second stationary value for the charge. Another means of increasing this limiting charge was afforded by the removal of the air surrounding the electrode. To show this the gauze cap was removed from the apparatus in fig. 2, and the metal tube siurounding the wire was brought forward and sealed to the anode of the discharge tube. The arrangement is shown in fig. 3, With this apparatus it was found that, as the exhaustion proceeded in the chamber B, the negative charge received by the electrode A gradually increased, until finally, at a very high vacuum, a momentary discharge of the rays was sufficient to raise its potential beyond the range of the electrometer. This result, therefore, confirms the explanation already given of the discharging action of the rays. In a recent paper by LENARDt this charging action of the cathode rays in a high vacuum was described, but its connection with the ionized air surrounding the electrode was not brought out. From the experiments just described it is clear that, while this action is directly due to the fact that the cathode rays carry a negative charge, the extent of the effect obtained in all cases depends to a very great degree upon the opposing influence exerted by the ionized air surrounding the electrode upon which the rays fall. 4. Ionization not due to Rontgen Rays. It has been thought by some that the ionization pixxluced by c«.thode rays was due to Rontgen rays, which might possibly be sent out from the window at the same time. The results of experiment are, however, entirely opposed to this view. * *Phil. Mftg.,' Noveml>er, 1897, p. 436. t * Wied. Ann.,' vol. 63, p. 253. Digitized by Google 56 MR. J. c. Mclennan on electrical conductivity in gases In order to investigate the point an apparatus similar to that shown in fig. 3 was adopted. Different thicknesses of aluminium foil were in turn used for the window, and the air in the chamber B was kept at a pressure low enough to absorb but little of any radiation coming from the window, and yet sufficiently high to aflford considerable conductivity when ionized. With diflferent thicknesses of the foil down to "04 millim., it was found that the electrode A did not gain any charge when the tube was excited. Further, if in these cases a charge, either positive or negative, was given independently to the electrode, this charge was maintained when the discharge passed in the tube, and no leak occurred. But when the window was made of foil '008 millim. in thickness, the effect obtained was such as that already described in the last paragraph. Under these conditions the electrode A, if carrying initially a positive or a negative charge, finally assumed a stationary state, in which it carried a definite negative charge whose value, as has already been pointed out, depended upon the pressure of the air in the chamber B. As, then, no leak from the electrode occurred when the aluminium was '04 millim. in thickness, it seems justifiable to conclude that if any Rontgen rays were present under these circumstances they were of an extremely weak character. If Rontgen rays of even very moderate intensity had entered the chamber, a leak would have taken place which could have been observed. In practice the aluminium foil used in my experiments was about '008 millim. in thickness, and with this foil intense ionization was observed. From the known character of Rontgen rays, it was quite impossible for this great ionization to be produced by rays which could be absorbed by a layer of aluminium "032 millim. — ^the difference in thickness of the two windows. Again, an ordinary focus tube illustrates very well the fact that the Rontgen rays produced issue in a large measure firom the face of the anticathode, upon which the cathode rays fall, while the radiation appearing to come fi:'om the opposite face is always very weak. The theory now generally accepted is that the Rontgen rays are electromagnetic pulses sent through the ether when the moving electrified particles which constitute the cathode rays are suddenly stopped. If then the Rontgen radiation sent out in the direction of propagation of the cathode rays, when these carriers were stopped by foil '04 millim. in thickness, was at most but very feeble, it appears highly improbable that a strong radiation of this kind could be produced by those carriers that passed through the thinner foil without being stopped. The conductivity produced in a gas by cathode rays is, moreover, far in excess of that excited by even the strongest Rontgen rays. In order to make a direct com- parison, measurements were taken of the ionizations produced in the same chamber by both radiations, and the following illustration gives an indication of their respective efficiencies. By using the apparatus shown in fig. 1, it was found that, under the action of cathode rays with a saturating intensity of field, a capacity of Digitized by Google THAVEBSED BY CATHODE RAYS. 57 750 electrostatic units attached to the electrode B gained in 15 seconds a charge represented by 300 divisions on an arbitrary scale. A Rontgen ray focus tube giving out very strong rays was then used in place of that for producing the cathode rays, and was excited by an induction coil capable of giving a 50-centims. spark. Under these circumstances, with the same field, which was also in this case a saturating one, a capacity of 150 electrostatic units was charged in one minute to an amount represented by 20 on the same scale. This case, which is an extreme one, shows that the ionization by cathode rays was about 300 times that due to an intense Rontgen radiation. In the present investigation these latter rays, even if they did accompany the cathode rays, must have been very feeble, and could there- fore only exert an ionizing influence which may be left out of consideration. The known action of a magnetic field naturally suggested itself as a means of sifting out the cathode from any accompanying Rontgen rays. The intensity of the cathode rays, however, soon falls off owing to their rapid absorption by the air, and on this account it was necessary to place the chamber in which the ionization was measured close up to the discharge tube. Under these conditions it was found impossible to deflect the rays outside the tube without also deflecting those inside. This difficulty consequently rendered the test indecisive, and the method had to be abandoned. 5. Discussion of Methods for Measuring the Ionizations produced in Different Gases. In the construction of Rontgen-ray bulbs, the disengagement of gas from the electrodes and the inside of the glass is facilitated by the application of heat to the tube. In the case of Lenard tubes, however, the joints are made of wax, and the final stage of exhaustion cannot be hastened by adopting this device. In practice a tube was kept attached to the mercury pump, and exhausted while the discharge was passing through it. After some hours of this procedure the coil was stopped, and the exhaustion was continued until only some traces of air were being taken over. On then exciting the tube, the vacuum was found to be sufficiently high for the cathode rays produced to penetrate the aluminium window. After running the coil for a short time, a small quantity of gas acciunulated in the tube, and the pressure rose so high that the rays ceased to be propagated outside. After this air had been removed the vacuum again became good, and the original intensity of tlie rays was restored. As the ionizing power of the rays was very great, charges sufficiently large to be accurately measured were easily accumulated by exciting the tube only for short periods. By following this course quite satisfactory results were obtained and much loss of time was avoided. On account of this running down of the discharge tube, it was impossible, in comparing the ionizations in two different gases, to use an apparatus with a single chamber, such as that shown in fig. 1. In order to obtain accurate results, it was VOL. CXCV. — A. I Digitized by Google 58 MR. J. c. McLennan on electrical conductivity in gases necessary either to have a constant source of rays, or else to be able to ascertain the relative intensities of the rays used with the different gases. One method which suggested itself was the use in series of two chambers, such as that shown in fig. 1. By inserting a thin aluminium membrane between them a different gas could be put in each chamber, and a single pencil of rays could be used to produce the ionization in both chambers. With this arrangement it was thought that the ionization obtained in the first chamber might perhaps bear a constant ratio to that produced in the second. But this relation was not found to hold, and further, as the cathode rays are rapidly absorbed, the amount of ionization obtained in the first chamber was so very much greater than that in the second, that even if the ratio had been fairly constant the method would not have been at all satisfactory. This led to a trial of two receivers in parallel. Although the cathode rays on issuing from the window diverge very greatly, mechanical difficulties made it im- practicable to receive part of the issuing rays in each chamber, and so recourse was had to the use of two windows. With a single large disc as cathode, a stream of rays was received in each of the chambers. The ratio of their intensities, however, as measured by the ionizations they produced, did not remain constant but varied quite irregularly. The explanation of this is probably found in a paper by A. A. C. SwiNTON,* where he points out that the carriers are shot off in a hollow cone from the cathode, and that the dimensions of such a cone of rays vary with the degree oi exhaustion in the tube. Besides, the aluminium windows were opposite to eccentric points on the cathode, and the ratio of the intensities of the two pencils was in this way greatly influenced by slight variations in the directions of the rays within the tube. A cathode formed of two small discs was then tried, and the results obtained were very satisfactory. The ratio of the discharges from the windows was in this case quite constant, and it was therefore possible to make measurements with con- fidence. The main difficulty of the investigation was in this way overcome, and the method was applied to obtain among other things a knowledge of — 1, the absorption of the rays ; 2, the ionizations produced by them in air at different pressures ; and 3, the relative ionizations in different gases. 6. Description of Apparatus used. A diagram of the apparatus is shown in fig. 4, and the way in which the connec- tions were made is exhibited in fig. 5. The exciting tube was slightly over 3 centims. in diameter. The two discs of the cathode were each about a centimetre in diameter, and they were placed with their centres directly in front of the alimiinium windows. That portion of the apparatus in which the ionizations were * *Proc. Roy. Soc.,' vol. 61, p. 79 (1897). Digitized by Google TRAVERSED BY CATHODE RAYS. 59 measured consisted of two chambers, A and B, each made of brass and similar in form to that shown in fig. 1. The two electrodes C and D were held in position by ebonite plugs, which closed the ends of the receivers and at the same time served as insulators. el*^ $arf^ In each experiment the receivers themselves were well earthed, and also, initially, the electrodes C and D. As the electrostatic induction was very intense in the neighbourhood of the discharge tube, it was found necessary to take special precautions in regard to the earth connections. Wires of but very small resist- ance were used, and these were led to water mains and all the joints carefully soldered. The two chambers were separated by a disc of ebonite, and to its faces were attached thin brass plates, a and h. By means of wires passing out through the I 2 Digitized by Google 60 MR. J. c. Mclennan on electrical conductiyity in gases ebonite to the battery of storage cells E, these plates could be charged to any desired potential, high or low. As the electrodes and the walls of the receiver were earthed, this afforded a means of setting up in each chamber a field which could be readily modified. The fields themselves, moreover, were quite distinct, each disc serving as a screen to cut off any action arising fi'om the other. Each of the chambers was provided with a projecting shoulder, which slid over a corresponding one on the anode surrounding the window opposite. By coating these joints with wax the chambers were then not only made airtight, but also were entirely separated from each other. In the apparatus used, the diameter of the chambers A and B was about 3 centims., and the distance between each of the electrodes and its corresponding plate a or h about 1'6 centims. The diameter of the narrow cylinders which admitted the rays to the chambers was 3 millims., and the distance between the aluminium windows and points corresponding to the centres of the electric fields was about 2 centims. \>XNXX\XXX\XX\\\\\\\\\\\\N>X\\X\ Each of the electrodes was connected to an air condenser, whose capacity was about 600 electrostatic units. These condensers, G and H, were each made of two sets of parallel plates separated by small ebonite supports. The plates were made by coating both .sides of a sheet of glass by a single sheet of tinfoil. In this way plates tolerably plane were obtained, and yet difficulties arising from electric absorption were avoided, the glass merely serving as a support for the foil plates. The measurements were made with a quadrant electrometer F, and the tube was excited by a 50-centims. spark-length induction coil, whose positive terminal, together with the anode of the tube, was kept to earth. This coil was provided with an Apps interruptor, and besides being very powerful was also very efficient. It Digitized by Google TRAVEESED BY CATHODE RAYS. 61 required a potential of only eight volts to excite it, and with the interruptor working slowly, this was suflScient to produce sparks of the maximum length in air at normal pressure. In practice, the interruptions were made at the rate of 20 to 25 per second. 7. Explanation of the Method adopted foi* Comparing Ionizations. It is well known that, in conduction in Rontgenised gases, and in gases acted upon by uranium radiation, the current of electricity obtained does not increase in propor- tion to the electromotive force applied. The current, after reaching a certain critical value, becomes practically stationary, and increases but very little when very large increases are made in the electromotive forces. This maximiun, or saturation current, was also found to characterise the conductivity produced by the passage of cathode rays through a gas. With Rontgen or uranium radiation, a field of 400 or 500 volts a centimetre has been found to give saturation in most simple gases; but with cathode rays it was necessary to apply fields of much stronger intensity. As already stated, the distance between either of the electrodes C and D, fig. 5, and the dividing partition was about 1 '6 centims. In order to ascertain the saturating electromotive force, the plate h was kept at a very high potential, while that of a was gradually increased fi'om zero. At each stage the ratio of the currents obtained in the two chambers was noted, and it was not until a potential of about 900 volts was applied to a that an approximation to the saturation current was obtained in the chamber A. With a potential difference of 1200 volts the increase in the current was small, and an increase only slightly larger was obtained with a potential of 1600 volts, or 1000 volts a centimetre. This small increment in the current very probably arose from the influence of the field itself. It may be that in certain parts of the receiver the rays, acting in conjunction with the applied differ- ence of potential, had not quite sufl&cient intensity to produce dissociation. An increase in the field under these circumstances would produce greater ionization, and consequently a larger current would be obtained. As this field of 1000 volts a centi- metre practically produced saturation currents in both chambers, it was used through- out in measuring the ionizations. Sparking was prevented by using in the charging circuit liquid resistances, such as xylol. An explanation of the saturation current is that the number of ions used up by the current in a given time is exactly equal to the number .produced by the rays in the same time, or in other words, the ions are removed so rapidly by the applied field that recombination is practically eliminated. The saturation current is then a direct measure of the ionization produced, and in order to compare the ionizations in any two gases, it suffices to measure their saturation currents. In this investigation the saturating electromotive force was applied to the plates a and 6, the discharge tube was then excited, and the currents obtained were used to charge up the con- Digitized by Google 62 MR J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES densers G and H. The discharge having been stopped, the potentials of the two condensers were then successively determined. As the effective capacity of the electrometer was the same fraction of that of each of the two equal condensers, the deflection readings were direct measures of the charges obtained. The charging of both condensers proceeded for the same time, and consequently the electrometer deflections were also direct measures of the saturation currents, and therefore of the ionizations in the two chambers. The method possessed the advantage of being independent of the time of charging and of the strengths of the rays coming from the two windows, provided only that the ratio of their intensities remained constant. In using the electrometer the needle was kept at a high potential, and one pair of quadrants always connected to earth. Though with this arrangement slow losses from the needle occurred, yet the short interval required for the two readings made the gradual change in the effective capacity of the electrometer inappreciable. In practice, the electrometer was initially connected to one of the condensers, and the tube allowed to run until a suitable deflection was obtained. After noting this reading, the electrometer, having been put to earth, was then connected to the other condenser and the second reading taken. In this way the ratio of the ionizations in the two chambers was obtained. From the experiments described in Section 2, it is clear that the signs of the charges obtained in the condensers depended on the signs of the charges given to a and h by the battery. In case these plates were positively charged, the charges collected were positive, and were due entirely to ionization. With a nega- tive field, however, the negative charges obtained included not only negative ions produced by the rays, but also the negative carriers, constituting the rays, that were stopped in their motion by the gas. For this reason the positive field was always used, and consequently the charges obtained gave a measure of the number of ions produced in the gas by the passage of the rays. 8. Ionization in different Gases at the Same Pressure. To compare the ionization in a selected gas with that in air at the same pressure, the saturating electromotive force was applied to the plates a and &, fig. 5. The two chambers A and B were first filled with air at atmospheric pressure, and a series of readings taken, the mean of which gave the ratio of the saturation currents in the two chambers. The air was then removed from A, and the gas to be tested introduced. A set of readings similarly taken gave a ratio for the saturation current obtained with the given gas in A, compared with that obtained with air in B. The combination of these results gave the ratio of the saturation current in A, when filled with the given gas, to that in the same chamber when filled with air. This ratio was, consequently, the ratio of the ionization produced in the selected gas Digitized by Google TRAVERSED BY CATHODE RAYS. to that produced in air at an equal pressure under the action of cathode rays entering the chamber with the same intensity in both cases. The results obtained from this method for hydrogen, air, and carbon dioxide are given in the first column of Table I. In the second column are given the relative ionizations found by J. J. Thomson* for these same gases when ionized by Rontgen rays of constant intensity. Table I. Name of gas. Column I. Column 11. Ionization by cathode rays. Ionization by Rontgen rays. Hydrogen Air Carbon dioxide . . . 2-65 1-00 •34 •33 100 1-40 These numbers, it will be seen, present a very marked difference. In the one case the ionization decreased as the density of the gas traversed increased, while in the other a law directly the reverse of this was followed. One explanation of this difference in the results is that the character of the ionization under cathode rays may be essentially different from that produced by Ilontgen rays. Apart from these numbers, however, there seems to be but little ground for this view. Strong experimental evidence now exists to support the assumption that the cathode rays consist of small particles of matter carrying nega- tive charges of electricity. We may therefore regard the ionization they produce as being due to their impinging on the molecules of a gas, and to the consequent breaking up of the latter. On this hypothesis it is not clear that the resulting ions should differ in character from those produced under the influence of Rontgen radiation. It appeared rather that the true explanation was to be found in the varying absorbing powers of the different gases. LENARD,t who studied these rays by the fluorescence they excited, found that the absorption of cathode rays by gases at atmospheric pressure was considerable. He was also led by his experiments to propound the law, that while different gases at the same pressure absorbed the rays to different degrees, yet their absorption depended only upon the densities of the gases, and not upon their chemical composition. In the apparatus here used, the distance traversed by the rays after they left the discharge tube until they reached the centre of the field where the ionization was ♦ *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12, t • Wied. Ann.,* vol. 56, p. 265 (1895). Digitized by Google 64 MR. J. c. Mclennan on electrical conductivity in gases measured, was about 2 centims. From Lenard's conclusions, it is obvious that in this distance the absorption of the rays by carbon dioxide would be greater than by air, and very much greater than by hydrogen. The effective intensities of the rays in the three gases at the same pressure would then be very diflferent, and numbers such as those given in Column I. follow naturally under these circumstances, without assuming any difierence in the character of the two ionizations. 9. Ionization in Air at Different Pressures. In order to study more closely the influence of absorption, a number of experiments were carried out similar to that just described. The same apparatus was used, and the same method followed, but the ionizations, instead of being measured in different gases at the same pressure, were determined for the same gas at different pressures. Table IL Pressure. Ionization measured. millims. 767 1-00 530 1-44 340 1-92 205 2-32 104 2-68 53 2-74 Between 40 and 45 millims. a sudden large increase was obtained in the ionization. This was found to be due to the action of the field itself in dissociating the gas. The results obtained with air are shown in Table II. The pressures are expressed in heights of columns of mercury at the same temperature. The ionizations given are relative, that corresponding to atmospheric pressure being taken as unity, and each value is the average of a large number of readings. The results are also shown graphically in fig. 6, where the abscissae represent pressures, and the ordinates corresponding relative ionizations. The numbers show that as the pressure decreased the ionization obtained with a saturating electromotive force steadily increased, until a pressure of about 75 millims. of mercury was reached. This result, though surprising, can be readily explained by the great absorption of the rays at atmospheric pressure. The rays had to travel at least 1 "5 centims* from the window before they reached that part of the chamber from which the saturation current was obtained. For this Digitized by VnOOQ iC TRAVEliSED BY CATHODE RAYS. 65 reason their efTective intensity was very largely determined by the pressure of the gas traversed. While a diminution in the pressure would not affect the original intensity of a pencil of rays issuing from the window, it would, owing to a decrease in the absorp- tion, increase the ionizing power of this pencil at the centre of the receiver. In this way, although the available amount of matter to be ionized was lessened by lowering the pressure, it could happen that the resultant ionization, as measured by the satu- ration current, would at first exhibit increasing values. This in all probability accounts for the numbers obtained in Table H. Now frora this point of view such a condition would only hold down to a stage when the two influences produced equal effects. The ionization would then be a maximum, and would afterwards fall off with diminishing pressures. Although the numbers obtained for the saturation current do not show definitely that a maximum value was obtained for the ionization, still there are indications from them, as the curve shown on fig. 6 illustrates, that the maximum value was reached at a pressure of about 75 millims. of mercury. /on/zat/ofTS 4 3 ^^ .-i^ 2 / ^.^ ^ \ lOO ?( 90 fi 00 5 10 m H> — 70 "T 55" A W i } PRsaauRcs IN MMS. Fig. VI. As indicated in Table II., the conditions of the experiment made it impossible to measure the ionization in air at pressures much below 50 millims. At about 40 millims. pressure a sudden large increase was obtained in the value of the satura- tion currrent, whch was found to be due to the influence exerted by the applied field in breaking down the gas. At these low pressures the electric intensity, which was 1000 volts a centimetre, wassufiicient to dissociate the attenuated gas and to produce a discharge on its own account between the electrodes. This was shown by simply connecting the electrometer to one of the electrodes, C for example, and applying the potential difference without exciting the discharge tube. On then exhausting the VOL. CXCV. — A. Digitized by Google 66 MR. J. c. McLennan on electrical conductivity in gases chamber, the electrometer showed no leak until the critical pressure was reached, when it inmiediately began to charge up. 10. Ionization in a Gas independent of its Chemical Composition. An important result in connection with these experiments is the agreement exhibited between the number given in Table I. for the ionization in hydrogen at atmospheric pressure and that given in Table II. for the ionization in air at a pressure of 53 miUims. Here two gases, hydrogen and air, were introduced in succession into the same measuring chamber and adjusted to the same density. Cathode rays of the same intensity were projected into this chamber in the two cases, and these rays, after traversing a certain length of the gas, reached a point where the ionization they there produced was measured. The values obtained show that under the circum- stances the same number of ions was produced in both gases. Since the rays issuing from the window were in both cases of the same intensity, it follows from Lenard's absorption law that the disposition of the rays, their actual intensities, and the quantities of them absorbed from point to point in the chamber, were precisely the same in both gases. Under these circumstances, therefore, the equal ionizations obtained in hydrogen and in air at the same density not only form a confirmation of Lenard's absorption law, but also show that where equal absorption occurs equal ionization is produced. In the case of Rontgen radiation, Rutherford* has made a determination of the relative absorbing powers of a number of gases. Taking I to denote the intensity of the rays on entering a particular gas, and le"^ their intensity after traversing a length X, he has found that the values of the coefficient of absorption for the different gases practically represented the relative conductivities produced in these same gases by Rontgen rays. It is thus interesting to note that with cathode rays, just as with Rontgen rays, equal absorption gives equal ionization. To test still further the accuracy of this conclusion a detailed examination was made of the ionization produced in a number of different gases. Throughout the experiments air in the chamber B, fig. 5, was taken as the standard. In some comparisons this air was kept at atmospheric pressure, while in others lower pressures were taken, the pressure selected being maintained through each complete determination. In making a comparison the chamber A was fiJled in turn with the two gases to be examined, and their pressures were adjusted so as to reduce them to the same density. Two ratios were in this way found for the ionizations in the chambers A and B, and as the influence of absorption was eliminated on account of the equal densities, these ratios represented the relative ionizations in the two gases under cathode rays of the same intensity. * * Phil. Mag.,' April, 1897, p. 254. Digitized by Google TRAVERSED BY CATHODE RAYS. 67 These ratios were determined by taking the mean of a number of readings. Samples of the results obtained in five different comparisons are given in Tables III., IV., v., VL, and VIL, the numbers being consecutive readings with each gas in A. They represent very well the working of the method. Although the variations were considerable, similar ones occurred in both sets of observations in each comparison, and as the number of readings taken was very large, any errors were in a great measure compensated. Table III. — Oxygen and Air. Air in both chambers at Oxygen in A at 675-1 millims. 746-7 millims. Air in B at 746-7 millims. Ionization in A. Ionization in B. Ionization in A. Ionization in B. 109 100 1-40 1-00 1-37 1-28 99 1-54 1-39 9 1-24 1-20 » 1-35 1-54 1 1-25 1-41 } 1-07 1-20 9 1-41 1-26 9 1-54 1-41 9 1-31 1-33 9 1-32 100 1-34 1-00 Table IV. — Nitrogen and Air. Air in A at 734-3 millims. Nitrogen in A at 757 millims. Air in B at 757 millims. Air in B at 757 millims. Ionization in A. Ionization in B. Ionization in A. Ionization in B. 1-10 1-00 1-04 1-00 1-02 99 M2 1-21 J J 1-03 1-05 J 9 1-34 1-29 i 1-03 110 9 106 1-18 9 M5 1-04 9 M2 1-07 9 1-08 1-00 n 1-06 Ml 1-00 MO 1-00 K 2 Digitized by Google 68 MR. J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES Table V. — Carbon Dioxide and Air. Air in both chambers at 772-7 millims. Carbon dioxide in A at 504-7 millims. Air in B at 772-7 millims. Ionization in A. Ionization in B. Ionization in A. Ionization in B. 1-22 100 1-17 1-00 1-12 )) 116 » 1-17 ) 1-23 9) 1-3S } 1-31 )) 1-02 » 1-37 }) 1-30 1 1-00 yj 111 y 1-21 )) 117 y 1-24 )) 103 f 1-31 yy 1-23 > 100 i» 1-17 1- 00 1-20 1-00 Table VI.— Hydrogen and Air. Air in both chambers at Hydrogen in A at 770*9 millims. 53*2 millims. Air in B at 53*2. Ionization in A. Ionization in B. Ionization in A. Ionization in B. 1-58 1-00 1-52 100 1-77 1-82 }} 1-64 1-91 ) 1-41 1-63 y 1-62 1-58 y 1-63 1-80 y 1-79 1-70 y 1-73 1-32 y 1-85 1-75 y 1-81 2-04 > 1-68 1-00 1-71 100 Digitized by Google TRAVERSED BY CATHODE RAYS. 69 Table VII. — Nitrous Oxide and Air. Air in both chambers at 759 minima. Nitrous oxide in A at 499 millims. Air in B at 759 millims. Ionization in A. Ionization in B. Ionization in A. Ionization in B. 1-08 1-00 103 100 MO 1-21 99 1-24 M5 9) 112 1-07 )) •99 113 11 1-08 1-07 „ 1 111 117 99 1-23 1-02 99 1-07 1-05 )) 112 110 »> 111 1-00 1-10 100 Care was taken to insure the purity of the gases, and they were also well dried before being passed into the ionizing chamber. The oxygen was prepared electroly tically, and was freed from ozone by being passed through a strong solution of potassium iodide and caustic potash. The nitrogen was prepared by gently heating a mixture of ammonium chloride with a nearly saturated solution of sodium nitrite. The gas given off was passed through a U-tube containing strong caustic potash, and also through a second containing concentrated sulphuric acid. A Kipp apparatus was used for the preparation of carbon dioxide, which was made in the ordinary manner by allowing dilute hydrochloric acid to act on marble. In making hydrogen a Kipp apparatus was also used, dilute sulphuric acid being allowed to act on zinc. The gas was passed through a strong potassium permanganate solution, and then through a U-tube containing a strong solution of caustic potash. The nitrous oxide was prepared by heating ammonium nitrate in a flask, and the gas was collected over water. Digitized by Google 70 MR. J. c. McLennan on electrical conductivity in gases Table VIII. — Summary of Measurements. Gases compared. Pressures. Ionizations. 1 Air mean of 30 readings Oxygen „ 30 „ millim*. 746-7 675-1 1-31 1-32 Air mean of 25 readings Nitrogen .... » 25 „ 734-3 757 Ml 1-09 Air mean of 30 readings Carbon dioxide . . « 30 „ 772-7 505-4 1-20 118 Air mean of 18 readings Hydrogen .... „ 18 „ 53-2 770-9 1-70 1-79 Air mean of 23 readings Nitrous oxide . . i, 23 „ 759 499-3 1-09 1-10 A summary of complete sets of observations on the different gsises is given in Table VIII. This statement includes the number of readings made in each case and the pressures at which these were taken. The ionizations quoted are the averages of the several sets of readings. The close agreement exhibited by the numbers corresponding to each comparison fully bears out the conclusion deduced from the earlier experiments. It not only forms a striking corroboration of Lenard's absorption law, but also shows that the ionization follows an analogous one, which may be stated thus : — When cathode rays of a given strength pass through a gas, the number of ions produced per second in 1 cub. centim. depends only upon the density of the gas, and is independent of its chemical composition. The similarity in the laws of absorption and ionization, holding, as it does, with so many gases over such a wide range of pressures, is a clear indication that when cathode rays are absorbed to a certain extent, the positive and negative ions produced by these absorbed rays are of a definite amount, which bears a constant ratio to the quantity of the rays absorbed ; that is to say, the absorption of a definite amount of radiant energy is always accompanied by the appearance of a fixed amount of potential energy in the form of free ions. This granted, it follows that in order to ascertain the relative ionizations pro- duced in any two gases by cathode rays of the same intensity, it is sufficient to determine the absorbing powers of the two gases for the same rays. In other words, Digitized by Google TRAVERSED BY CATHODE RAYS. 71 the coefficients of ionization for a series of gases are fully determined when the coefficients of absorption for these same gases are known. The existence of this general relation between absorption and ionization for both cathode and Rontgen rays is especially interesting when we remember that the two radiations are so very different in many respects. In the one case, according to the generally-accepted view, the rays consist of small charged particles of matter moving with high velocities in space, while in the other they are supposed to consist of electromagnetic impulses propagated in the ether. With the one the dissociation is in all probability brought about by a series of impacts between the moving particles and the molecules of the gas ; with the other it seems to be due to the direct action of the intense electric field forming the impulse. Again, while the absorption of cathode rays depends only upon the density of the medium traversed, the absorption of Rontgen rays, according to Rutherford's results, does not seem to depend to any great extent upon the molecular weight of the gas. But while all these differences exist in the two radiations, with both of them it holds good that the same number of ions are always produced in a gas when the same amount of rays traversing it are absorbed. 11. Comparison of Ionizations produced hy Cathode and hy Rontgen Rays. The method just described gives definite and conclusive information regarding the ionizations produced by cathode rays in gases of the same density ; but where the gases are of different densities, it cannot be satisfactorily applied. As stated in Section IX., the rays, after entering the ionizing chamber, must travel some distance before reaching that part of the field fi'om which the current is drawn. On this account, though rays entering the chamber may originally be of the same strength, still their effective intensities become at ordinary pressures quite different, when the gases traversed are not of the same density. Also as it is impossible to define exactly the disposition of the electric field within the chamber, these effective intensities cannot be calculated with any degree of accuracy. A difficulty arises, too, from the dispersion of the rays. As shown by Lexard, they issue from the window in a pencil whose form is greatly influenced by the density of the gas traversed. At very low pressures they pass through the aluminium window practically without deviation, but as the pressure increases, they spread out until finally they issue in all directions. The conclusion arrived at in the last section, however, suggests a means of calcu- lating the ionization which would be produced by rays of constant intensity in different gases at the same pressure. Lenard,* who investigated the absorption powers of a number of gases at different * * Wied. Ann.,' vol. 56, p. 258. Digitized by Google 72 MR. J. c. mclen:nan on electrical conductivity in gases pressures, has shown that for any particular gas the coefficient of absorption varies directly as the pressure. In the case of air, taking I to denote the intensity of the rays issuing from the window of the discharge tube, and le"^ their intensity at a distance x from the window, he found for \ the values given in Table IX. Table IX. Air pressure • Coefficient of absorption. millims. 760 3-43 331 1-51 165 •661 83-7 •396 40-5 •235 19-3 •117 100 •0400 2-7 •0166 •78 •00416 These numbers, it wiU be seen, amply support Lenard's conclusion. Similar tables, given by him for a number of gases, all exhibit the same relation between the values of X and the corresponding pressures of the gas. Now, if the values of the coefficient of absorption are taken to represent the rela- tive ionizations produced in a gas, at a point where the pressure is varied but the intensity of the rays kept constant, it follows from Lenard's numbers that the ionization in any particular gas would vary directly as the pressure to which it was subjected. This result, which follows as a deduction from the preceding experiments, has also been found experimentally by Perrin* to characterise the ionization produced by Rontgen rays. It is true that with Rontgen rays a number of experimenters have found quite different relations to hold between the ionization and the pressure ; but in most cases they have vitiated their results either through omitting to use satu- rating electromotive forces, or through neglecting to arrange their experiments so as to eliminate the metal effect observed by Perrin. With uranium radiation also, RuTHERFORDt has found the ionization to be propor- tional to the pressure of the gas traversed. The direct experimental verification of a law of this kind is always accompanied by a serious difficulty. The law has reference to the action of rays whose intensity is constant throughout the region ionized. With rays that are easily absorbed by gases at ordinary pressures, this condition can be realised either by the use of very thin layers of gas or by investigating the ionizations at very] low pressures. Owing * *Comptes Rendus,' vol, 123, p. 878. t *Phil. Mag.,' January, 1899, p. 136. Digitized by VnOOQ iC TRAVERSED BY CATHODE RAYS. 73 to mechanical difficulties, however, the former method is generally impracticable, while the action of the applied electric field in breaking down the insulation of the gas precludes the use of the latter artifice. It is then open to measure the ionizations produced by rays, traversing layers of gas of considerable thickness. But before any relation connecting ionizations and pressures can be deduced from such measurements, it is necessary to have definite information regarding the absorptive powers of the gases at different pressures, and to know exactly the form and dimensions of the region from which the ions are drawn. Although the absorption laws for cathode rays have been fully developed by Lenard, and are quite definite and clear, it is scarcely possible to define even approximately the region in the ionizing chambers (fig. 5) from which the ions go to make up the saturation current. On this account a direct verification of the proportionality law is not possible ; but, as already pointed out, the results of the experiments described in Section X. strongly support the conclusion that, in the case of a gas subjected to increasing pressure, the ionizations produced by rays of constant intensity bear the same ratio to each other as the coefficients of absorption corresponding to these pressures. If, then, the ionization in a gas varies with the pressure, it follows at once that if rays of the same intensity were allowed to traverse thin layers of different gases at a constant pressure, the ionizations produced would be directly proportional to the densities of these gases. Take, for example, carbon dioxide and air. It has been shown that the ionization produced in carbon dioxide at a pressure of 5047 millims. of mercury is the same as that produced in air at 7727 millims. by rays of the same intensity. According to the proportion law the ionization produced by these same rays in CO2 at 772 7 would then be just r53 times that obtained at the lower pressure ; that is, with rays of the same intensity the ionizations in carbon dioxide and in air would be to each other as 1*53 to 1 when these gases were subjected to the same pressure. A similar conclusion may be deduced from a consideration of the other gases examined. Hence, on this view, the relative ionizations produced by rays of constant intensity in a series of gases subjected to the same pressure would be expressed by the numbers which under these circumstances give their relative densities. These numbers are given for the gases examined in Column I., Table X., while in Column II. are given the values found by J. J. Thomson* for the relative ionizations produced by Rontgen rays of constant intensity in the same gases. * *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12. VOL. CXCV. — A. L Digitized by Google 74 MR. J. C, McLENNAU on electrical CONDUCTIVITY IN GASES Table X. Column I. ' Column XL Gases examined. Densities (shown above to be proportional to ionization by cathode rays), air = 1. Ionization by Rontgen rays. Ionization of air taken as unity. Air Oxygen Nitrogen .... Carbon dioxide . . Hydrogen .... Nitrous oxide. . . 1-00 1-00 1-106 M •97 -89 1-53 j 1-4 •069 •as 1-52 1-4.7 The numbers, with the exception of those for hydrogen, present an agreement which is very striking, and show that although the two forms of radiation are so very different, still the products of their action upon the gases cited are practically the same. While the difference in the nimibers for hydrogen is very large, there seems to be some doubt as to the proper value to be assigned to the conductivity produced by Rontgen rays in this gas. The conductivities under Rontgen rays in the gases named have been measured by a number of experimenters, and while their values for the other gases differ but little, a very wide divergence exists in their numbers for hydrogen. Rutherford* gives the value '5, while PERRiNf bas obtained the number '026 by a method entirely different from that of any of the others. Though we have been thus led to conclude that the density of a gas should determine its conductivity under cathode rays, strong evidence exists against adopt- ing any such general conclusion regarding the conductivity produced by Rontgen rays, notwithstanding the general agreement indicated above for the gases cited. With such gases as HCl, Cl^, SO2, and HgS, J. J. Thomson, Rutherford, and Perrin have found the conductivities given in Table XI. From an examination of these values and a comparison with those of Table X., it is evident that it is quite impossible to deduce any such relation between the densities of the gases and their conductivities under this radiation. * * Phil. Mag.,' April, 1897, p. 254 t * Th^se pr^ent^ k la Faculty des Sciences de Paris,* 1897, p. 46. Digitized by Google TRAVERSED BY CATHODE RAYS. 75 Table XL — Conductivity under Rontgen Rays. Gas. Deiisitv. Measurcil by Pkrrin. J. J. Thomson. Eithkrford. I • HCl 1-25 .8-9 11 SO., : 2-23 6*4 4 i Cl.> 2-45 17-4 : 18 , U^S 119 CO 6 8 : G I Although the laws of ionization and absorption for cathode rays are clearly defined by these results, it is difficult to apply them in practice to the direct calculation of the relative ionizations in any particular experiment. Take, for example, the case of a pencil of pai*allel rays, 1 sq. centim. in cross section, traversing air at a pressure p. Let q = the rate at which ions are produced in 1 cub. centim. of air at unit pressure by cathode rays of unit intensity and \) = the coefficient of absorption of air for unit pressure. Consider then the ionization between two planes distant x and x + c/x, from the source of the rays. If I denotes the original intensity of the rays, I . e"**^ will represent their intensity at a distance x, and p.q .1. c'^^'^'dx wiU then represent the total number of ions produced between these two planes in one second. Imagine now a saturating electric field applied at right angles to the rays and confined between the limits r and r -{• d. The value of the total saturation current obtained with this field would then be fr+d p.q.l. e'^^dxy or i = ^.e-^'-(l-^e-^^) (1), At) where pX^ is replaced by the quantity X, whose values for different i)ressuro8 are given in Table IX. If the air traversed be now subjected to diminishing pressures, the saturation current will assume different values and will reach a maximum when i.e., {r+ d)e-'"'^r = 0, .- = '• + '' (2). L 2 Digitized by VjOOQ IC 76 MR. J. c. Mclennan on electrical conductivity in gases An experiment somewhat analogous to this is described, in Section IX. The apparatus used is shown in figs. 4 and 5. The diameters of the electrodes C and D were each about 1 centim. and, as already stated, the distance between the window and the centre of each of the chambers was about 2 centims. By applying the equation (2) to this experiment, and taking r = 15 centims.* and d = I centim., it follows that the saturation current would be a maximum when €^= 1-66 ... or X = '5. From Lenard's values, Table IX., it will be seen that this value corresponds approximately to a pressure of about 120 millims. of mercury. The observed results, however, Table II. and fig. 6, indicate a maximum of about 75 centims. Further, the calculated values of the current from equation (1) exhibit a more rapid rise than that actually observed. But the difierence in the results is not surprising. The field within the receiver was far from uniform, being disturbed by the proximity of the walls of the chamber. The presence of the narrow tube through which the rays were conducted into the receiver also produced irregularities. On this account it was impossible to define, even approximately, the region from which the saturation current was drawn. Moreover, the actual paths of the rays, as Lenard has pointed out, are largely influenced by the pressure of the gas traversed. Even at best, then, the calculated results can scarcely be regarded as more than a rough approximation. 12. Summary of Results, 1. The conductivity impressed upon a gas by cathode rays is similar to that produced by Rontgen and uranium rays, and can be fully explained on the hypothesis that positive and negative ions are produced by the radiation throughout the volume of the gas traversed. 2. When cathode rays are allowed to fall upon insulated metallic . conductors surrounded by air at atmospheric pressure, (a.) such conductors if initially uncharged gain a small limiting negative charge, (6.) positive charges are completely dissipated, (c.) negative charges drop to a small limiting value, (o?.) the loss of charge is due to the action of the ionized air surrounding the conductor, and the value of tlie limiting negative charge is determined by the extent of the conduction in this air. 3. The ionization produced in a gas by rays coming from the aluminium window in a Lenard discharge tube is due to cathode rays and not to Rontgen raya Digitized by Google TRAVERSED BY CATHODE RAYS. 77 4. Lenard's results obtained by fluoroscopic methods on the absorption of cathode rays are confirmed by a study of the ionization these rays produce in gases. 5. When cathode rays of a given strength are passed through a gas, the number of ions produced in 1 cub. centim. depends only upon the density of the gas, and is independent of its chemical composition. 6. With rays of constant intensity the ionization in any particular gas varies directly with the pressure to which it is subjected. 7. The relative ionizations produced by cathode rays of constant intensity in air, oxygen, nitrogen, carbon dioxide, hydrogen, and nitrous oxide, at the same pressure, are expressed by the numbers which represent their densities. 8. With cathode rays, just as with Rontgen rays, the number of ions produced in a gas bears a definite ratio to the amount of the radiant energy absorbed. I gladly avail myself of this opportunity to record my grateful sense of the never failing encouragement and assistance received from Professor J. J. Thomson. Digitized by Google Digitized by Google [ 79 ] III. Mathematical Contributions to the Theory of Evolution. — VIII. On the Inheri- tance of Characters not capable of Exact Quantitative Measurement. — Part I. Introductory. Part II. On the Inheritance of Coat-colour in Horses. Part III. On the Inhentance of Eye-colour in Man. By Karl Pearson, F.R.S., ivitk the assistance of Alice Lee, D.Sc.y University College^ London. Received August 5, — Read November 16, 1899 ; withdrawn, rewritten, and again received March 5, 1900. Contents. Pakt I.— Introductory. Page § 1. General Nature of the Problem and Assumptions upon which it can be solved 80 § 2. Determination of the Mean Value of the Characters and the Ratio of the Variability of Correlated Characters 81 § 3. Determination of the Probable Errors and Error Correlations of all the quantities involved . 82 § 4. On the Construction of Normal Scales for Characters not capable a prim'i of exact quantitative Measurement 87 § 5. On Blended and Exclusive Inheritances 88 Part II. — On Coat-colour Inheritance in tlie Tlwrougldned Horse. § 6. On the Extraction and Reduction of the Data 92 § 7. On the Mean Coat-colour of Thoroughbred Horses 93 § 8. On the Relative Variability of Sex and Generation 94 § 9. On the Inheritance of Colour in Thoroughbred Horses 98 (a) Direct Line, First Degree, {b) Direct Line, Second Degree. {c) Collateral Inheritance. General Conclusions. Part III. — On Eye-colour Inheritance in Man. § 10. On the Extraction and Reduction of the Data 102 §11. On the Mean Colour, having regard to Sex and Generation 104 § 12. On the Relative Varial)ility of Sex and Generation 109 §13. On the Inheritance of Eye-colour in Man 113 (a) Aflsortative Mating, (b) Collateral Heredity, First Degree, (c) Collateral Heredity,. Second Degree, (d) Direct Heredity, First Degree, (e) Direct Heredity, Second Degree. (/) On Exclusive Inheritance. § 14. General Conclusions 119 Appendix I. Tables of Coat-colour in Horses 122 II. Tables of Eye-colour in Man 138 Note I. Inheritance of Temper and Artistic Instinct 147 Note 11. On the Correlation of Fertility and Eye-colour 148 VOL. CXCV. — A 264. 29.10.1900 Digitized by Google 80 PROFESSOR K. PEARSON AND DR. A. LEE ON NOTE. Thifl memoir was originally presented to the Society on August 5, 1899, and read on November 16, 1899. In working out by the same theory the coefficients of inheritance for Basset Hounds, Mr. Leslie Bramley-Moore discovered that the method adopted was not exact enough in its process of propor- tioning. Accordingly, with the assistance of Mr. L. N. G. Filon, we immensely developed the theory, so that it was necessary to rewrite the theoretical part of the original memoir. This has been carried out in Part VIL of this series. The present memoir consists substantially of the portions of the original memoir relating to the inheritance of coat-colour in Horses and eye-colour in Man, with the numerical details and the resulting conclusions modified, so far as the extended theory rendered this necessary. In the very laborious work of reconstructing my original tables I have received the greatest possible assistance from Dr. Alice Lee, and I now wish to associate her name with mine on the memoir.* The memoir was at my request returned to me for revision after it had been accepted for the * Philosophical Transactions.* Part I. — Introductory. (1.) A CERTAIN number of characters in living forms are capable of easy observation, and thus are in themselves suitable for observation, but they do not admit of an exact quantitative measurement, or only admit of this with very great labour. The object of the present paper is to illustrate a method by which the correlation of such characters may be effectively dealt with in a considerable number of cases. The con- ditions requisite are the following : — (i.) The characters should admit of a quantitative order, although it may be impossible to give a numerical value to the character in any individual. Thus it is impossible at present to give a quantitative value to a brown, a bay, or a roan horse, but it is not impossible to put them in order of relative darkness of shade. Or, again, we see that a blue eye is lighter than a hazel one, although we cannot d priori determine their relative positions numerically on a quantitative scale. Even in the markings on the wings of butterflies or moths, where it might be indefinitely laborious to count the scales, some half dozen or dozen specimens may be taken to fix a quantitative order, and all other specimens may be grouped by inspection in the intervals so determined. We can even go a stage further and group men or beasts into simply two categories — ^light and dark, tall and short, dolichocephalic and brachycephalic — and so we might ascertain by the method adopted whether there is, for example, correla- tion between complexion and stature, or stature and cephalic index. (ii.) We assume that the characters are a function of some variable, which, if we * I have further to thank Mr. Leslie Bramley-Moore, Mr. L. N. G. Filon, M.A., Mr. W. R. Macdonell, M.A., LL.D. and Miss C. D. Fawcett, B.Sc, for much help in the arithmetic, often for laborious calculations by processes and on tables, which were none the less of service if they were afterwards discarded for others. To Mr. Bramley-Moore I owe the extraction and part of the arithmetical reduction of the horse-colour tables. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 81 could determine a quantitative scale, would give a distribution obeying — at any rate to a first approximation — the normal law of frequency. The whole of the theoretical investigations are given in a separate memoir, in which the method applied is illustrated by numerical examples taken from inheri- tance of eye-colour in man, of coat-colour in horses and dogs, and from other fields. We shall not therefore in this paper consider the processes involved, but we may make one or two remarks on the justification for their use. If we take a problem like that of coat-colour in horses, it is by no means difficult to construct an order of intensity of shade. The variable on which it depends may be the amount of a certain pigment in the hair, or the relative amounts of two pigments. Much the same applies to eye-colour. In both cases we may fail to obtain a true quantitative scale, but we may reasonably argue that, if we could find the quantity of pigment, we should be able to form a continuous curve of frequency. We make the assump- tion that this curve — to at any rate a first approximation — is a normal curve. Now if we take any line parallel to the axis of frequency and dividing the curve, we divide the total frequency into two classes, which, so long as there is a quantitative order of tint or colour, will have their relative frequency unchanged, however we, in our ignorance of the fundamental variable, distort its scale. For example, if we classify horses into bay and darker, chestnut and lighter, we have a division which is quite independent of the quantitative range we may give to black, brown, bay, chestnut, roan, grey, &c. Precisely the same thing occurs with eye-colour ; we classify into brown and darker, hazel and lighter, and the numbers in these classes will not change with the quantitative scale ultimately given to the various eye-tints. Our problem thus reduces to the following one : Given two classes of one variable, and two classes of a second variable correlated with it, deduce the value of the correlation. Classify sire and foal into bay and darker, chestnut and lighter ; mother and daughter into brown and darker, hazel and lighter, and then find the correlation due to inheritance between the coat-colour or eye-colour of these pairs of relations. The method of doing this is given in Memoir VII. of this series. Its legitimacy depends on the assumptions (i.) and (ii.) made above, which may I think be looked upon as justifiable approximations to the truth. Of course the probable error of the method is larger than we find it to be when cor- relation is determined from the product-moment. Its value varies with the inequality of the firequency in the two classes given by the arbitrary division. It will be least when we make that frequency as nearly equal as possible — a result which can often be approximately reached by a proper classification. In our present data the probable errors vary from about "02 to '04, values which by no means hinder us from drawing general conclusions, and which allow of quite satisfactory general resulta (2.) So far we have only spoken of the two classes, which are necessary if we merely want to determine the correlation. But if we wish to deal with relative VOL, CXCV. — A, M Digitized by Google 82 PEOPESSOR K. PEAESON AND DR. A. LEE ON variability we must have more than two classes. We have, in fact, in our tables preserved Mr. Galton's eight eye-colour classes and the seventeen classes under which the coat-colour of thoroughbred horses is classified in Wethbrby^s studbooks. Such a classification enables us at any rate approximately to ascertain relative variability, and, what is more, to reconstruct approximately the quantitative scale according to which the tints must be distributed in order that the frequency should be normal. For, in order to attain this result, we have to ascertain from a table of the areas of the normal curve the ratio of the length of the abscissa to the standard deviation which corresponds to any given increase of frequency. Let us suppose that three classes have been made — n^, n^, nj, represented by the areas of the normal curve in the accompanying diagram so marked. Let pi and p^ be the distances of the mean from Ox, x^ the two boundaries of n^ Here Pi may be negative, or p^ infinite, &c. Then if Aj = p^ja-y A3 = pj(r^ we find at once, if N = total frequency, "^^ = viiy""^ (>■)• "■'r-'" = a/!D-"<^ (-)• Now the integrals on the right are tabulated, and thus, since the left-hand side is a known numerical quantity, it follows that pja and Ps/o-, and accordingly the range (Ps ~V\)I^ ^^ *^® (AsiSR in terms of the standard deviation, are fully determined. Thus, if € be the range on the scale of tint or colour of the group of which the observed frequency is ng, we have € = j^g — p^, and thus c/cr = q say, is known. For a second series c/cr' = q\ Hence a-ja = q/q, and accordingly the ratio of the variabilities of the two series is determined. Again, the ratio pj{pz — Pi) enables us to find the position of the mean in terms of the range on the scale occupied by the tint corresponding to the frequency n^. As a rule we shall take this tint to be that in which the mean actually lies, in which case we shall have pjipz + P\) as determining the ratio in which the mean divides the true quantitative range of this particular tint. (3.) Let 7; = p^{pz-Pi) = h/{K-K) (iii-)> ^ = cr/ar' = (V-V)/(^3-^) (iv.)- It remains to find the probable errors of these quantities. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 83 Suppose Sj. to be the standard deviation for the errors in a quantity a?, and R,y the correlation coefficient for errors in two quantities x and y. Further let H^;^'-" W' where subscripts and dashes may be attached to H to correspond to like distin- guishing marks attached to h. Since 2^1-^ _ 1 te-i^dx hi ) 2N -v/2^Jo*' "^ ^^^-'^ we have at once Snj =: NHjS^i, and S,. = 2«y(NH0 (vU.). Similarly, Sn^ = — NH38A3, whence : 24. = W(NH3) (viii.). Further, we have 2«.2>i,R*.«, = - S,,2..R«.,y(N«HiH3) (ix.) ; but, as is shewn in Part VII., § 4, , __ Wi(N-«A ^ ss _ n,(N-n,) , . MA..= -^ • . . (^.). Thus we find Probable error of ^1 = *67449Si, _ -67449 1_ A(N-n,) , •• . ~ v/N Hi V N2~~ ^^"•^• Probable error of A3 = 'T/W'W A/ "' n~ (xiii.). S Correlation in errors in hi and A3, or R|^^, is given by 2*.S^RiA=l^ (^^^-^ Let u = A3 — Aj, tt' = A3' — A/ be the ratio to the respective standard deviations of the ranges corresponding to the groups n^ and rij'. Then ~ N»l Hi« "^ H,« HiH,J ' whence, if i/ be a proportional frequency = n/N, we readily find Prol»bleerrorofu = -^''{^+i-(^_ + ^7 . . . (xv,). Rotable «ror of u'=:^'{|l+|l-(i + ^,)*}'. . . (.vi.). M 2 Digitized by VjOOQ IC 84 PROFESSOR K. PEARSON AND DR. A. LEE ON I now proceed to determine the correlation in the errors made in determining the ranges corresponding to any two classes of any two variables which are correlated. For this purpose let the frequency correlation table be dressed as follows, in the diagram below. kA-j>i< — -h.-k- » i Axis of X variable, g ': '/ *. TotAL ^ §• /77„ m^i "^Sl n', \s-0- M ^ \ ^%y, > q'' If- m,g m^g fUss "i > 1 1 T s 5& V^ HsHJ) fn,s /»« Mas r>^ ToCclLs ", n. n^ N Here m,y denotes the frequency of individuals common to the two classes n,- and riy Let M(/ denote its " conjugate," or all the frequency which appears in neither Ui nor v!j ; then N = My + n, + n'j — m^ (xviL). As before, we have J ^ n i(N — n,) N 2_ »t'XN-«0) N g _ 'MyCN-^fy) ^ 2 _ mi,(N - V/iy) '^ — N ' ^^ — N (xviii.). (xix.). Further, since m-^ and M(; are mutually exclusive, we have Mj/ZWy From (xvii.) we have for small variations (XX.). Hence 2S^X,R.^,= V+V-S-.*- V-22.,tH,RM^ . . . (xxi). Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 85 Substituting the values given above we find, after some reductions, 2„.2«..K^,;,^ = —^-^ ' (xxu.) This result, which is extremely simple in form, gives the correlation in errors made in determining the frequencies in any two classes whatever of any two correlated variables. I next proceed to find the correlation between errors in u and u\ the ratio of the ranges occupied by any two classes to their respective standard deviations. We have 8n^ = — NH38A3. Hence S(A, - *,) = ^ + I'd; - i). Similarly 8(V - A,-) = ^. + f (g. - |.). Multiply the first by the second, and summing as usual for all possible errors, we have, by using (xxii.) ^^p _ 1 JNmaa - 71^71^' . Nm^g -' n^n^' / 1 1 \ 'Sm^ - n^\ / 1 1\ z«z,.Xi,^ - N 1 N^HiH/ " ■•" N»Hi VHi' sj ""^ N^H/ \Ei, ~ hJ Collecting the like H's we find, after very considerable reductions, . . . . (xxiiL), , Z.2,^n,,. - jj I jj^jj^, -h jj^jj^, -f jj^g^, -I- jj^jj^, I . . vxxui. ;. or, where /ty = wi^/N = proportional frequency. A glance at our diagram on the previous page of the correlation table divided into nine classes, shows at once the symmetrical formation of this result. By writing at the points P, Q, S, and T, the ordinate there of the normal surface, on the supposition of no correlation and N = 1, the construction of the result is still more clearly brought out. We are now in a position to determine the probable errors of 17 and ^ We have 8,,= __ h^Bh^ — AjS/ij Digitized by VjOOQ IC 86 PROFESSOR K. PEARSON AND DR. A. LEE ON Hence - «*N 1 (AiH,)«Na "^ (A8H3)'N2 (A,Hi)(A2H,)N2i ' Or, Probable error of t) where u is the range A3 — h^, and v^ and 1^3 are the proportional frequencies, as before. Care must be taken, if the class n^ cover, as it usually will in our present investiga- tions, the mean, to put h^ negative within the radical. In other words, for a class covering the mean we have Probable error of rj - '67449 h,K r_?L . _-s_ _ fjH Ys^yV /xxv^ Lastly we have u* u \u' tt / * *f ~ »» \ V,'* "^ tt« ««' J • Thus : Probable error of ^ = -67«.i{|'+^--^}' (.xvi.), where we have by (xv.), (xvi.), and (xxiii."*) 2 8-1 Jiii . jv _/< . ivyi V < p _ i J /*ii - "i^i' , M »8 - ^sV , /*i8 - yiV _i_ /^i r "V!!'! *"^"' """Nl H,Hi' ^"HsH,' "•■ H1H3' +'H3Hi' /' where, as before, fi'a and v'a represent proportional frequencies. In the following investigations on coat-colour, and eye-colour inheritance I have not thought it needful to give in every one of the thirty-six relationships dealt with the probable errors of the means, ratio of variabilities, and the coefficients of inheri- temce (»;, 4> and r). The arithmetical labour would have been too great, for the or, Hence Digitized by VjOOQ IC MATHEMATICAL CONTEIBUTIONS TO THE THEORY OP EVOLUTION. 87 expressions as given above are somewhat complex. It is, however, necessary to keep the approximate values of these probable errors in view, and, as our results classify themselves easily into groups for which our data, as well as the intensity of heredity, are approximately the same, one series of these errors has been found for each group. (4.) If we have ground for our assumption that the variable at the basis of our tint classification can be so selected as to give a normal distribution, we may deter- mine the relative lengths on the scale of that variable occupied by each tint or shade. Thus if (Ti be the standard deviation of the variable for male eye-colour, o-g for female eye-colour, I measured the range on the scale in terms of o-j and a^ for Mr. Galton's eight eye-colour tints for 3000 cases of male and 3000 cases of female eye-colour. I found the spaces occupied on the unknown scale to be as follows : — No. Tint. Eange in terms of 0-2. Range in terms of o-i. 1 2 3 4 5 6 7 8 Light blue . * Blue, dark blue Grey, blue-green Dark grey, hazel Light brown . Brown . . . Dark brown. . Very dark brown. black! 00 1-39276 •73468 •40027 •03893 •43679 •84161 00 00 1-34918 •77596 •41992 •00856 •35895 •64167 00 These results are not so regular as we might have hoped for, on the assumption that the ratio of a-i/a-t^ would be the same from whatever part of the scale it be determined. The general conclusion, however, would be that a-^ is slightly larger than 0*2, which is confirmed by other investigations. Actually a tint may be rather vaguely described, and where the data were obtained by untrained observers without the assistance of a plate of eye-colours, a good deal of rather rough classification is likely to have taken place. I do not think it would be safe to go further than stating that on the quantitative colour scale the tints as described must occupy spaces in about the following proportions : — Light Blue. Blue, Dark Blue. Grey, Blue-Green. Dark Grey, Hazel. Light Brown. Brown. Dark Brown. Very dark Brown, Black. 00 1-37 •75 •41 •02 •40 •74 00 Taking 2000 colts and 2000 fillies, the standard deviations being a-i and 0-3 respec- tively, I have worked out the coat-colour ranges in terms of 0-3 and cr, for each of the sixteen colours* occurring in the records. We have the following results ; — * See p. 92, below. Digitized by Google 88 PROFESSOR K. PEARSON AND DR. A. LEE ON Tint. Range in 0-2. Range in a-^. Range in o-j. Range in o-j. 1 00 ao 9 •00000 •00000 2 •12683 •10768 10 1-96956 2-01658 3 •00000 •03313 11 •00000 •00000 4 •91747 11 1055 12 •02490 •00000 5 •00000 •00352 13 •00000 •00000 6 •11059 •10451 14 •00000 •00000 7 1 •34684 1 •27688 15 •00000 •00000 8 •00000 •00000 16 00 00 Here again it seems to me that the most we can safely do is to consider that on a suitable scale the relative lengths occupied by the classes of coat-colours recognised by thoroughbred horse breeders would be somewhat as follows : — bl. bl./hr. br./bl. br. br./b. b./br. b. b./ch. ch./b. ch. ch./ro. ro./ch. i ro. ro/gr. gr/ro. g^. ' 00 •12 •02 roi •00 •11 1^31 •00 ' •oo 1-99 •00 •01 •00 •00 •00 00 1 The reader must carefully bear in mind that these represent scale-lengths occupied by the coat-colour and not the frequency of horses of these individual coat-colours. What we are to understand is this : that if eye-colour in man and coat-colour in horses were measured on such quantitative scales as we have given in skeleton, then the distribution of the frequency of the several colours would be very approximately normal. The actual skeleton scales are represented in the accompanying diagram, which puts them at once before the eye. br/bC. T Normal Scale 0(f Colour Ranges in Thoroughbred Horoes. tibr. ch/h. BLsLck Brown Bay Chestnut smn tr/b. ^m chM'Si^. Normal Scale of Eye Colour Ranges in Man. am^^^"^' Brtmr, %S^ Grey BLue-6reen Odrk blue. Blue LighC BlSe (5.) It is necessary here to draw attention to a distinction of some importance in heredity, namely, that between blended and exclusive inheritance. In my treatment of the law of ancestral heredity,* it is assumed that we have to deal with a quanti- tatively measurable character, and that the ancestry contribute to the quantity of this character in certain proportions which on the average are fixed and follow certain definite numerical laws. Such an inheritance is blended inheritance. But another * *Ro7/Soc. Proc.,' vol. 62, p. 386. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 89 type of inheritance is possible. We may have one progenitor, prepotent over all others and absorbing all their shares, who hands down to the of&pring not a proportion of his character, but the whole of it without blend. If this progenitor is a parent we have exclusive inheritance, if a higher ancestor a case of reversion. I have dealt at some length with this type of inheritance imder the title of the Law of Reversion in another paper.* We must consider in outline the main features of such inheritance, for the cases of eye-colour in man and coat-colour in the horse approximate more closely to the numerical values required by it, than to those indicated by the law of ancestral heredity. The chief feature of exclusive inheritance is the absolute prepotency of one parent with regard to some organ or character. It need not always be the parent of the same sex, or the same parent throughout the same family. Some offspring may take absolutely after one, others after another parent for this or that organ or character only. I believe Mr. Galton first drew attention, in his * Natural Inheritance' (p. 139), to this exclusive or, as he terms it, alternative heritage in eye-colour. In going through his data again I have been extremely impressed by it ; even those cases in which children might be described as a blend, rare as they are, are quite possibly the result of reversion rather than blending. If we suppose exclu- sive inheritance to be absolute, and there to be no blending or reversion, it is not hard to determine the laws of inheritance. Supposing the population stable, one-half the oflGspring of parentages with one parent of given eye-colour would be identical with that parent in eye-colour, the other half would regress to the general population mean, i.e., the mean eye-colour of all parents. Hence, taken as a whole, the regression of children on the parent would be '5. In the case of the grandparent the regression would be '25 ; of a great grandparent '125, and so on. With an imcle a quarter of the oi&pring of his brother will be identical in eye-colour with him, the other three- quarters will regress to the population mean, thus the regression will be '25. If we have n brethren in a family, and take all possible pairs of fraternal relations out of it, there will be \n{n — 1) such pairs ; ^ brothers will have the same eye-colour that of one parent, the other \n brother that of the other parent. Hence selecting any one brother, \n — 1 would have his eye-colour, and on the average \n would have regressed to the mean of the general population. In other words, the coefficient of regression would be {\n — l)/(Jn -^ 1 + ^) = (^n — l)/(w — 1). Accordingly «= 3 Begressiou = -25 n = 4 = -3333 «= 4-7 = -3649 » = 5 = -375 n = 5-3 = -3833 n = 6 = '4 n = 00 = '5 ♦ *Koy. JSoc. Proc./ vol. 66, pp. 140 d setj. VOL. CXCV. — A. N Digitized by Google 90 PROFESSOR K. PEARSON AND DR. A. LEE ON It will at first appear, therefore, that the fraternal regression with the size of families actually occurring will vary from '35 to '4. To some extent these values would be modified by assortative mating, which actually exists in the case of eye-colour. The correlations between parent and offspring and between brothers would both be slightly increased. Thus if p be the coefficient of assortative mating, Ry the fraternal correlation with and Vf without assortative mating, and r the coefficient for parent and offspring,* If we put Vf = '36, ?' = '5, /o = '1, we find R/ = -39. Thus we see that the regression or correlation for fraternal inheritance in the case of exclusive inheritance could not, with the average size of families, be very far from '4 of blended inheritance. A further source which can modify immensely, however, the fraternal correlation is the prepotency of one or other parent, not universally, but within the individual family. In the extreme case all the offspring might be alike in each individual family. Thus fraternal correlation might be perfect although parental correlation were no greater than -5. Hence, where for small families we get a fraternal correlation greater than •4 to '5, it is highly probable that there exists either a sex prepotency (in this case, one of the parental correlations will be considerably greater than the other) or an individual prepotency (in which case the parental correlations based on the average may be equal). We shall see that fraternal correlations occur greater than '5 in our present investigations. I have dealt with these points in my Memoir on the * Law of Reversion,'! and also in the second edition of the ' Grammar of Science. 'J Another point also deserves notice, namely, that with the series '5, '25, '125, &c., for the ancestral coefficients in the direct line, the theorems proved in my Memoir on Regression, Heredity, and Panmixia§ for the series of coefficients ?% r^, r^ . . . exactly hold. That is to say, if we have absolutely exclusive inheritance, the partial regres- sion coefficients for direct ancestry are all zero except in the case of the parents. This it will be observed is not in agreement with Mr. Galton's views as expressed in Chapter VIII. of the * Natural Inheritance.' But I do not see how it is possible to treat exclusive inheritance on the hypothesis that the parental regression is about '3.11 Actual investigation shows that for this very character, i.e., eye-colour, it is nearer '5. If we take Table XIX. of Mr. Galton's appendix, and make the following groups, both * This is shown in a paper not yet published on the inj9uence of selection on correlation. + *Roy. Soc. Proc./ vol. 66, pp. UO et seq„ t " On Prepotency," p. 459 ; " On Exclusive Inheritance," p. 486. § ^Phil. Trans.,' A, vol. 161, p. 302, etc. II Mr. Galton takes I throughout his arithmetic. Digitized by Google MATHEMATICAL C50NTRIBUTI0NS TO THE THEORY OF EVOLUTION. 91 parents light, one parent light and one medium, one light and one dark, we reach the following results : — Parents* eye-colour. Children, actual. Light-eyed children, calculated. Total. l-'o-Hy^i.: iJSJl Ancestral law with knowledge of parents and grandparents. Both light. . . . Light and medium . Light and dark . . 355 215 211 334 355 170 161 107 105 1 321 160 117 1 Here the exclusive inheritance leads us to misplace thirty-two and the -ancestral law thirty-three children. The evidence, therefore, of the correctness of the latter is hardly greater than that of the former. Indeed, if the former were modified for reversion, it would very possibly give better results than the latter. I am inclined accordingly to look upon eye-colour inheritance as an exclusive inheritance modified by reversion, and, to some extent, by assortative mating, rather than a mixture of exclusive inheritance with a slight amount of blending. In either case exclusive inheritance gives results like the above so closely in accord with the ancestral law that the latter might be supposed to hold. But, theoretically, I do not understand how the ancestral law is compatible with exclusive inheritance. Theoretically, we have values of parental, avuncular, and grand-parental correlation greater than the ancestral law would permit of, and these theoretical values are, on the whole, closer to observation, as we shall see in the sequel, than those given by the law of ancestral heredity. The following table gives the two systems : — Table I. — Theoretical Values of the Regression Coefiicients. Belationship. Blended inheritance, ancestral law. Exclusive inheritance, absolute, no reversion. Parent and offspring Qrandparent and offspring . . . Great-grandparent and offspring . Brethren Uncle and nephew •3 •15 •075 •4 •15 •5 •25 •125 •36 to -5* •25 Now, if exclusive inheritance be modified by reversion or assortative mating, or if blended inheritance be modified by " taxation," t then we shall get values. different * This varies with the size of the family, t * Roy. Soc. Proc.,' vol. 62, p. 402. N 2 Digitized by VnOOQ iC 92 PROFESSOR K. PEARSON AND DR. A. LEE OK from the above, and possibly filling up the numerical gap between them. To this point I shall return when dealing with the observed values for eye-colour in man. Part II. — On Colour-Inheritance in Thoroughbred Racehorses. (6.) All the data were extracted from Weatherby's stud-books, the colours being those of the horses as yearlings. My first step was to form an order, not a quantita- tive scale, of horse-colours. With this end in view, the recorded colours were examined, and, including the arabs, the following seventeen colours were at first found : — 1. Black (bl.). 2. Black or brown (bl./br.). 3. Brown or black (br./bl.). 4. Brown (br.). 5. Brown or bay (br./b.). 6. Bay or brown (b./br.). 7. Bay (b.). 8. Bay or chestnut (b./ch.). 9. Chestnut or bay (ch./b.). 10. Chestnut (ch.). 11. Chestnut or roan (ch./ro.). 12. Roan or chestnut (ro./ch.). 13. Boan (ro.). 14. Roan or grey (ro./g.). 15. Grey or roan (g./ro.). 16. Grey(g.). 17. White (w.). Now, if we take the alternative colours to mean that the first alternative is the prominent element, we see that these colours in use among breeders admit of only one arrangement from black to white. That is to say, that a continuous shade-change is actually in use for the colour-nomenclature of thoroughbred horses.* Thus without any hypothesis as to the quantitative relative values of bay or roan, we have an order which serves for all our present purposes. Following this order. Appendix I., Tables I. — XII., for the colour correlation of related pairs of horses was compiled by Mr. Leslie Bramley-Moore from the stud-books. When dealing with relationship m the ? line only, no weight has been given to fertility, as each mare has had only one foal attributed to it, or two in the case of fraternal correlation. In the case of the c? line, the colours of the older sires were harder to ascertain, and we did not obtain altogether more than 600 sire-colours. Thus one, two, or, in a few cases, three or four colts or fillies were taken from each sire. I shall now discuss the results which may be drawn from these tables for the theory of heredity, first placing in a single table all the numerical constants calculated from the data in Tables I. to XII. of Appendix I. ^ Among the 6000--8000 horses dealt with only four were found with colours not entered in this scale, but these entries of bl./ch., br./ch., b./ro., in no way contradict the order of the scale, but merely show a rougher appreciation on the part of the nomendator, or possibly printers' or editor's errors. Digitized by Google MATHEMATICAL CONTBIBUTIONS TO THE THEORY OF EVOLUTION. 93 Table II. — Coat-colour Inheritance in Thoroughbred Horses. Pair of relatives. Division of bay range by the mean Ratio of variabilities. Coeffi- cients of correlar tion. Coefficients of regression. Num- ber of cases. X. y- Vx- Vv C = <ryl<rg. rxg. Rw Rv«. N. Sire Colt. . FiUy . Colt. . Filly . •6111 •6061 •5359 •5565 •5713 •5719 •6027 •6051 •8712 •8298 •9500 •9036 1-1478 1-2051 1^0526 11067 •4913 •5422 •4862 •5668 •4280 •4499 •4619 •5122 •5639 •6534 •5118 •6273 1300 1050 I 1000 ! 1000 Sire Dam Dam Maternal Maternal grandsire grandsire Colt. . Filly . •6583 •6359 •5867 •6042 •7030 •7678 r4225 13024 •3590 •3116 •2524 •2392 ■5107 •4058 1000 1000 Colt . Colt . Filly . Filly . Filly . FiUy . '(Half (Whole (Half (Whole (Half (Whole Colt. . siblings) Colt, . siblings) Filly . siblings) Filly . siblings) Colt. . siblings) Colt. . siblings) •5908 •5620 •5665 •5684 •5633 •5410 •5908 •5620 •6665 •5684 •5865 •5711 1 1 1 1 •9607 •9555 1 1 1 1 1^0409 1^0466 •3551 •6232 •4265 •6928 •2834 •6827 •3551 •6232 •4265 •6928 •2723 •5568 •3551 •6232 •4265 •6928 •2960 10466 2000 2000 2000 2000 1000 1000 In this table R,y = r^^^cxlcy, 'Ryx = ^jryO-y/cr^r. Half-siblings* are those having the same dam, but different sires. Further, 17 is measured from the brown end of the bay range up to the mean. (7.) On the Mean Coat-Colour of Horses. — If our theory be correct, this colour will not differ much from the median colour, and we notice at once a secular change going on. We have the following order : — Maternal grandsire of colt t; = '6583 Maternal grandsire of filly = '6359 Sire of colt = -6111 Sire of filly = 6061 Colt (mean value of seven series) . . . = '5816 Dam of colts t; = '5359 Dam of fillies = -5565 Fillies (mean value of seven series) . . = *575S * I have introduced this expression in my paper on "The Law of Reversion," *Roy. Soc. Proc./ vol. 66, as a convenient expression for a pair of offspring from same parents whatever be their sex. Digitized by VnOOQ iC 94 PROFESSOR K. PEARSON AND DR. A. LEE ON Now the colours of all the horses are returned when they are foals, so that there is no question of any variation of colour with age, yet we notice that — (i.) The horse is lighter in colour than the mare. (ii.) If we go back two generations (grandsire) the horse is lighter than if we only go back one generation (sire), and the sires are again lighter than their colts. In other words, there seems a progressive change towards a darker coat. (iii.) On the other hand, the mares one generation back appear to be darker than their daughters. (iv.) The average sire of colts is lighter than the average sire of fiUies ; the average dam of colts is darker than the average dam of fillies. Now these conclusions seem to indicate that the older horse was lighter in coat, and the older mare darker in coat than either the colt or filly of to-day, and that there is a tendency in the thoroughbred racehorse of to-day to approach to an equality of colour in the two sexes, an equality which is a blend of the sensibly divergent sex-colour of the older generation. Whether this secular change is due to the " breeding out " of the influence of light Arabian sires, or to a tendency in the past to select light-coloured sires and dark- coloured mares for breeding, or to the fact that such coloured sires and mares are the most fertile, i.e., to an indirect effect of reproductive selection, is not so easy to determine. But what does appear certain is that the average thoroughbred is approaching to a blend between its male and female ancestry, which were sensibly divergent.* (8.) On the Relative Variability of Sex and Class in Horses. — The following table gives the length of the bay range in terms of the standard deviation for each class. If c represent this range, then in terms of the previous notation c = w x <r = ?/ X o^, and from these values of it and it' the ratio, C = <^/<^' of Table II. was calculated. * Mean of dams and sires of colts = -5735, «.€., i(*6111 + -5359). Mean of dams and sires of fillies = *5813, i.e,, ^(*6061 + -5565). These are curiously enough almost exactly equal to the mean values '5753 and '5816 obtained for fillies and colts. This inverse relationship is too close to the probable errors of the quantities under investiga- tion for real stress to Ijc laid on it, but it may still turn out to be suggestive. Digitized by VnOOQ iC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 95 III. — Table of Bay Eanges. Relative Pair. Bay Ranga Probable Error of Median. X. y- U X a-,. u' X a-y. X. y- Sire Sire Dam Dam Colt .... Filly .... Colt .... Filly .... l-46943<r« l-64075<ra, l-36645<r, l-38165<ra l-28019(ry l-36I49<ry l-298I9<ry l-24845(ry ± 0160 ±•0159 ±•0196 ±0193 ± -0183 ± ^0192 ±•0206 ±•0214 Maternal grandsire . Maternal grandsire . Colt .... Filly .... l-69694<r, 1 -650210^, M9293<r„ l-26702<r„ ±•0158 ±0162 ± ^0224 ±0211 Colt (Half Colt (Whole Filly (Half Filly (Whole Filly (Half FUly (Whole Colt .... siblinin) Colt . . . . siblings) Filly . . siblings) Filly .... siblings) Colt .... siblings) Colt .... siblings) l-23953<ra, l-27688(r« l-39619(ra, l-34684<r, l-33479<r, 1 -415010^, l-23953<ry l-27688<ry l-39619<ry l-34684<ry l-28229<ry l-35207(r„ ±0153 ±•0148 ±0135 ±•0140 ±0202 ±•0189 ±•0153 ±0148 ±•0135 ±•0140 ±0208 ±0198 To explain the last double column I note that Mr. Shbppabd has shown (* Phil. Trans./ A, voL 192, p. 134) that the probable error of the median equals •84535 o-/\/N. Hence in terms of the bay range we have probable error of median .q^kqk// /m\ length of bay ranee /\ v /• length of bay range I have found that this simple result gives a value close to but slightly larger than the probable error of the quantity tj (p. 82), from which I have determined the position of the mean in the bay range. It is much easier to calculate, but of course not so exact, as we take no accoimt of possible errors in the bay range itself or their correlation with errors in the median. I have accordingly tabulated its values in the last double colunm as a rough guide to the errors made in the numbers upon which the statements in the previous section depend. I shall return to the consideration of the probable errors below. Turning to columns 3 and 4 of our Table 11. , we can draw the following conclusions as to the variability of sex and class: — (a.) The Younger Generation is more Variable than the Old. — Thus, foals are more variable than their sires, and, looking at the expressions in Table III. for the bay range, Digitized by VnOOQ iC 96 PROFESSOR K. PEARSON AND DR. A. LEE ON sii'es than grandsires. This is a rule I have now found true in a very great number of cases of inheritance. Parents are a fairly closely selected body of the general population, and so less variable than that population at large. This might appear pretty obvious in the case of thoroughbred horses when we are dealing with sires and grandsires. I have already pointed out that it was impossible to take 1000 to 1300 colts or fillies with as many independent sires, the fashion in sires is too marked ; and of course the number of independent grandsires was still fewer.* But even in the case of dams, where we have taken as many independent dams as fillies, we see this reduction in variability in the older generation. As it also occurs with stature, &c., in man as well as with coat-colour in horses — in which latter case we expect artificial selection — it deserves special consideration. Without weighting with fertility, there exists a selection of the individuals destined to be parents in each generation. We have to ask whether the change in mean and variability from parent to oflGspring in each generation is secular or periodic, or to what extent it is partly one and partly the other. The importance of settling this point is very great ; it concerns the stability of races. So far as my fairly numerous series of measurements yet go, I cannot say that a " stable population " has definitely shown itself; the characters of each race when measured for two generations seem to vary both in mean and standard deviation. Palseontologists tell us of species that have remained stable for thousands of years, but this is a judgment hitherto based on a qualitative apprecia- tion. A quantitative comparison of the means, variabilities, and correlations of some living species in its present and its fossil representatives would be of the greatest interest and value. For myself, I must confess that my nmnerical investigations so far tend to impress me with the unstable character of most populations. (6.) There is fairly good evidence that the horse is more variable than the 7na/re in coat-colour. It would be idle to argue fi:om the first four results of Table III. that the mare is more variable than the horse, in that these results show the dam to be more variable than the sire. For, as we have shown, the process of breeding and our method of extracting the data tend to produce a much more intense selection of sires than of dams. But if we compare the mean bay range in terms of the standard deviation of colts for our seven series of colts with that for the seven series of fillies in Table III., we find for the first 1*27458 <r<. and for the second 1 '33854 c/. Hence we are justified in concluding that a-c is greater than oy; In fact in only one case out of the seven does the series of fillies give a less variability than the corresponding series of colts, i.e., colts corresponding to dams are somewhat less variable than fillies corresponding to dams. It must, however, be remembered that this conclusion is based upon the coat-colour of the animals recorded as yearling foals, t Thus, the coat- * For some account of the extent of in and in breeding in the thoroughbred horse, see my memoir on " Reproductive Selection," • Phil. Trans.,' A, vol. 192, p. 267 et seq, t The reader must always bear in mind that when we speak of the variability of colour in sire or dam, <^c., it means the varialiility of this class when they were yearlings. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 97 colour may change both in intensity and variability with age, much as variability in stature changes with children from birth to adult life. (c.) As a more or less natural result of (6) it follows that any • group, male or female, having nutle relatives is more variable than the same group with female relatives. Thus sires of colts are more variable than sires of fillies ; fillies half-sisters to colts are more variable than fillies half-sisters to fillies, &c. But out of the nine cases provided by our data there are three exceptions to the rule, and perhaps not much stress can be laid on it, at iany rate in the above form. It would appear that males, relatives of males, are sensibly more variable than males relatives of females. The bay ranges are 1*3926 </ and 1*4447 a- respectively, which indicates that the average cr' is larger than a-. But if we treat the groups of females alone, we find for females with male relatives the bay range = 1*3694 <r, and for females with female relatives 1*3433 <t\ which indicates that the latter are more variable. The difference is, however, not very sensible. Possibly the rule is simply that extremes tend to produce their own sex, but our data are not sufficient for reaUy definite conclusions on the point. In order that we may have a fair appreciation of the probable errors of the quantities involved and the weight that is to be laid upon their differences, I place here a table* of the probable errors of ij, of £ = Cx/a-y and of r,^ for t3rpical cases. IV.— Table of Probable Errors. Belations. V' Vy C u u' ray Sire and Filly . . Grandsire and Colt Colt and Colt . . (Whole siblings) Filly and Colt . . (Half siblings) •0U3 •0143 •0186 •0179 •0170 •0199 •0186 •0185 •0243 •0237 •0315 •0363 •0385 •0328 •0335 •0330 •0319 •0328 •0328 •0288 •0333 •0259 ■0363 It will be seen from this table that the probable error in 17 is about 3 per cent., in C about 2 to 4 per cent., in u about 2 to 2*5 per cent., and the values of r about *03, growing somewhat larger as r grows smaller. The probable errors are thus some- what larger than those which we obtain by the old processes when the characters are capable of quantitative measurement, but they are not so large as to seriously affect the use of the new processes in biological investigations. As we have already indicated, the probable errors of the tj'b may be roughly judged by Mr. Sheppard's formula for the median (p. 95). It will be seen that the differences in the rfa and fs of Table II., or the u'q of * I have to thank Mr. W. R. Macponell for friendly aid in the somewhat laborious arithmetic involved in calculating these probable errors. VOL. CXCV. — A. O Digitized by VnOOQ iC 98 PROFESSOR K. PEARSON AND DR. A. LEE ON Table III., are as a rule larger than the probable errors of the diflferences, sometimes several times larger. Yet in some cases they are not such large multiples of the probable errors of the diflferences that we can aiSbrd to lay great stress on the divergence of ly or £ or tt in a pair of special cases. We must lay weight rather on the general tendency of the tables when all the series are taken together. Thus, while we may have small doubt about the correctness of (i.) of § 7 or (6) of § 8, we should look upon (iv.) of § 7 as an important suggestion which deserves serious consideration rather than a demonstrated law. The same again holds good for (c) of § 8. It is because of their suggestiveness that they are here included. That a differential fertility or an individualisation in the sex of offspring should be corre- lated with colour, would, if proved, be a fact of very considerable interest. It would again emphasise the important part which genetic selection plays in the modification of characters.* A priori it is not more unreasonable to expect coat-colour in horses than to suppose hair-colour in men to be correlated with fertility. But the fertility of man does seem to vary from the light to the dark races. The special feature of interest here, however, is that a different colour in the two sexes appears to influence the preponderance of one or other sex in the offspring. It would be an interesting inquiry to determine whether the sex-ratio in the oflfepring of " mixed marriages " varies when the races of the two parents are interchanged. (9.) On the Inheritance of Coat-colour in Thoroughbred Horses. — (a.) Direct Line. First Degree, — Having regard to the probable errors — ^about '03 — in the values of the correlation coefficient r^yy it seems quite reasonable to suppose that the mean parental correlation, actually '5216, is practically '5. It is quite impossible to imagine it the '3 of Mr. Galton's view of the Law of Ancestral Heredity. If we adopt the view of that law given in my paper on the Law of Ancestral Heredity, t and take the coefficient y to be different from unity, then it is shown in my paper on the Law of Reversion^ that we cannot on the theory of blended inheritance get parental correlation as high as '5 without values of the fraternal correlation which are much higher than those hitherto observed, certainly much higher than, as we shall see later, we find in the case of coat-colour in horses. Coat-colour in horses does not thus appear to be at all in accord with Mr. Galton's view of ancestral inheritance, or even with my generalised form of his theory. It does accord very well with what we might expect on the theory of exclusive inheritance as developed above, p. 91, on the assumption that there is no reversion. Looking at the matter from the relative standpoint, we see that not much stress can be laid on the respective influences of the sire and dam on the colt, or of the sire and dam on the filly ; but, on the other hand, the filly does appear to inherit more from * See the concluding remarks in the memoir on " Genetic (Reproductive) Selection," * Phil. Trans.,' A, vol. 192, pp. 257—330. t • Roy. Soc. Proc.,' vol. 62, p. 386 et seq. J 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 99 both parents than the colt does. There is certainly (judged from coat-colour) no preponderance of the sire's influence over the dam's such as breeders appear occasion- ally to imagine. The average influence of the dam on the ofispring indeed appears to be slightly greater than that of the sire, but the difference is of the order of the probable error, and not of the overwhelming character exhibited in the case of Basset Hounds. There is indeed in the case of thoroughbred horses not the same chance of carelessness produced by a misalliance afterwards screened by the defaulter. There exists, however, a far greater premium — considering the great value of yearlings from fashionable sires — set upon dishonesty. Again it is possible that when stallions receive too many public or private mares, or are still used in their old age, that they may, without losing the power of fertilising, lose some of the power of transmitting their character. The divergences, so far as the probable errors are concerned, are not such that we are forced out of our way to explain them. With the single exception of sire and colt we see that our table shows the universal prevalence of the rule that : Relatives of the same sex are more closely correlated than relatives of the same grades of the opposite sex. Thus : — A colt is more like his sire than his dam. A filly is more like her dam than her sire, A dam is more like her filly than her colt. A grandsire is more like his grand-colt than his grand-filly. A colt is more like his brother colt than his sister filly. A filly is more like her sister filly than her brother colt. the latter two cases being true for both whole and half siblings. The solitary exception is that a sire is more like his filly than his colt. If we were to assume it a rule that a filly in the matter of coat-colour has stronger inheritance all round than a colt, we should find it agree with our results for parental inheritance, and receive considerable support for the much stronger correlation of fillies than of colts, when either whole or half siblings. But it would not be in accordance with our results for grandparents, for which, however, we have only details for two out of the eight possible cases. On the whole, I think we must content ourselves with the statements that parental correlation is certainly about '5, and that with high probability each sex is more closely correlated with its own sex of the same grade of relationship. (6.) Direct Line, Second Degree. — My data here are unfortunately only for two cases out of. the possible eight. I hope some day to finish the series, but the labour of ascertaining from the studbooks the coat-colour of 700 or 800 separate sires is \rery great. Indeed it is not easy to foUow up the pedigree through the male line when the sire is not one of the famous few. On the other hand, it is much easier through the female line. For this reason the maternal grandsire was taken. Even O 2 Digitized by Google 100 PROFESSOR K. PEARSON AND DR, A. LEE ON in this case we had to seek back for each sire — the year of whose birth was unknown — ^until we found the record of his coat-colour given under the heading of his dam in the year of his birth. The average of our two cases gives a coefficient of correlation == '3353, the colt having a greater degree of resemblance to the grandsire than the filly. This value is substantially greater than the '25 we might expect for exclusive inheritance, and more than double the value '15, to be expected for the grandparental correlation with Mr. G Alton's unmodified law for blended inheritance. Of course the '25 is to be expected as the mean of the eight grandparental series, and, as we shall see for eye- colour in man, these may vary very much in magnitude. But allowing for this, it seems quite impossible that the average value could be reduced to '15. I take it therefore that the grandparental, like the parental, data point to a law of inheritance other than that which has been described in my paper on the Law of Ancestral Heredity. This peculiar strengthening of the grandparental heritage has already been noted by me in my paper on the Law of Reversion,* and the difficulties of dealing with it on the principle of reversion therein discussed. There may be some opinion among breeders as to the desirability of emphasising the dam's strain in the choice of a sire which leads to this result, but if so it is unknown to me, nor do I see how it would work without also emphasising the correlation of the dam and foaL The mean value of the correlation for the maternal grandfather and grandchildren for eye-colour in man is '3343 — a. result in capital agreement with that for coat-colour in horses. In that case the average of the eight series, as we shall see later, is con- siderably above '25, and we must, I think, suspend our judgment as to whether this could possibly in the case of horses be the true mean value. As to the value '15 it seems quite out of the question. As already remarked, the influence of the maternal grandsire (unlike that of the sire) is substantially greater on the colt than on the filly. (c.) Collateral Heredity, Fi7'st Degree. — Here we have more ample data to go upon, namely, a complete set of six tables of both whole and half siblings of both sexes. We notice one or two remarkable features straight off. In the first place, in the case of both fillies and colts, the whole siblings of the same sex have not a correlation the double of that of the half siblings, but have a correlation very considerably less than this. A priori we might very reasonably expect the one to be the double of the other. This is what would happen in the case of blended inheritance on the hypothesis of equipotency of the parents. As the half siblings are on the dam's side, we might assert a considerable prepotency of the dam over the sire. This cannot indeed be the explanation of the divergence in the case of Basset Hounds, where the half siblings have a correlation considerably less than half that of whole siblings, t * *Roy. Soc. Proc.,' vol. 66, p. 140 et seq. t * Roy. Soc. Proc.,' vol. 66, p. UO dseq. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 101 and yet the prepotency of the dam in coat-colour is very marked. But in the present case there is on the average only a slight, if indeed it be a real, prepotency of the dam. Further, if we turn to the correlation, no longer of siblings of the same sex, but of opposite sexes, we find the correlation of the whole siblings is approxi- mately double that of the half siblings, as we should d priori have expected. Taking averages on the assumption that the correlation for whole siblings should be double that for half siblings, we have the following results : — Correlation between colts based on results for whole and half siblings . '6667 Correlation between fillies based on results for whole and half siblings . '7729 Correlation between filly and colt based on results for whole and half siblings '5747 Mean correlation of siblings based upon all results for whole siblings . '6329 Mean correlation of siblings based upon all results for half siblings . . '7100 Mean correlation of siblings based upon results for both whole and half siblings '6714 We can draw the following conclusions : — (i.) In whatever manner we deduce the fraternal correlation it is very much larger than the '4 for whole brethren, or the '2 for half brethren, required by the unmodified Galtonian law. Such values, as we see above, could be deduced fi-om the modified Galtonian law by taking y greater than unity,* but this would involve values for the parental correlation sensibly less than those given by theory. We are again compelled to assert that the modified as well as the unmodified theory of blended inheritance, based on the Galtonian law, does not fit the facts. The above values, however, are quite compatible with the theory of exclusive inheritance on the assumption that there is an individual (not a sexual) prepotency from one pairing to another. (ii.) In whatever way we consider it, it would appear that the average value of the fraternal correlation, as deduced fi:om siblings with the same dam only, is greater than that deduced from siblings with both the same dam and the same sire. I am not able to explain this in any way, for I cannot assert a very substantial prepotency of the dam. All I can say from the data at present available is that for horses and dogs there appears to be no simple numerical relation between the correla- tion of whole and half brethren. (iii.) Offspring of the same sex are more alike than offspring of opposite sexes. This appears to be generally true, so far as our data at present extend, and will be fairly manifest fi^om the table below. * 'Boy. Soc. Proc.,' vol. 66, p. 140 e^ seq. Digitized by Google 102 PROFESSOR K. PEARSON AND DR. A. LEE ON Table V. — Collateral Heredity. 1 i Pair. ! Man. Dog. Horse. Stature.* Cephalic Index.! Eye-colour. J i Coat-colour.§ 1 Coat-colour. Whole Siblings. HaH Siblings. Brother-Brother . i Sister-Sister . . Brother-Sister . : •3913 •4436 •3754 •3790 •4890 •3400 •5169 •4463 •4615 i} -5257 1 •6232 •6928 •6827 •3561 •4265 •2834 It will be noted that, with the single exception of eye-colour in man, sister and sister are more alike than brother and brother. The mean value of the fraternal correlation for stature is '4034, and for cephalic index '4027. These results are in excellent accordance with the '4 required by the Galtonian theory of blended inheritance. The mean values for eye-colour in man, coat-colour in dogs, and coat-colour in horses are : '4749, '5257, and '6329. These are quite incompatible with that theory. I venture accordingly to suggest that we have here cases of an inheritance which does not blend, and that it is an inheritance which is far more closely described by the numbers we have obtained on the theory before developed of exclusive inheritance than by the law of ancestral heredity. Taking in conjunction with these results for collateral heredity, those for parental and grandparental inheritance, we see that coat-colour in horses diverges widely from the laws which have been found sufficient in the cases of stature and cephalic index in man. The latter characters are really based on a complex system of parts, while the determination of coat-colour may depend on a simple or even single factor in the plasmic mechanism. Here Mr. Galton's suggestion of an exclusive inheritance of separate parts ('Natural Inheritance,' p. 139) may enable us to understand why stature and cephalic index differ so widely in their laws of inheritance from coat- and eye-colours. Part HI. — On the Inheritance of Eye-Colour in Man. (10.) 071 the Extraction and Reduction of the Data. — The eye-colour data used in this memoir were most generously placed at my disposal by Mr. Francis Galton. They are contained in a manuscript wherein, by a simple notation, we can see at a * Pearson, * Phil. Trans.,' A, vol. 187, p. 253 et seq. See Note I. at the end of this paper. t Pawckti and Pearson, * Roy. Soc. Proc.,' vol. 62, p. 413 et seq, X Present memoir, p. IIS et seq, § Pearson, *Roy. Soc. Proc.,' vol 66, p. 140 et seq. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 103 glance the distribution in eye-colour of a whole family in its numerous male and female lines. Such complete details of the various direct and collateral relationships I have not hitherto come across, and from them I was able to form, in the course of some months of work, the twenty-four tables of correlation which are given in Appendix II. These tables, for the first time, give the whole eight series of grand - parental and the whole eight series of avuncular relationships, besides such as we have deduced for other characters previously, i.e., the four parental, the three fraternal rela- tionships, and the table for assortative mating. The very great importance of this material will at once be obvious, and I cannot sufficiently express my gratitude to Mr. Galton for allowing me to make free use of his valuable data. At the same time we must pay due regard to the limitations of this material, which it is needful to enumerate, so that too great stress may not be laid on the irregularities and divergences which arise when we attempt to reduce the results to laws. These limitations are as follows : — (a.) While the data of about 780 marriages are given in the record, they belong to less than 150 separate families. All our relationships, therefore, contain pairs weighted with the fertility of the individual families. Thus it was necessary to enter every child of a mother, every nephew of an uncle, and so forth. In the horse data we could take 1000 distinct mares and give to each one foal only. That is not possible in the present case. (6.) The colour of eyes alters considerably with age. It is not clear that some of the eye-colours are not given for infants under twelve months, and certainly the eye- colours in the case of grandparents and others must have been taken in old, perhaps extreme old, age. In a large number of other cases of great grandfather, great great- grandfather, &c., great uncles, and so forth, the eye-colours must have been given from memory or taken from portraits — in neither alternative very trustworthy sources. Thus while the horse colour is always given for the yearling foal by the breeder, the eye-colour is given at very different ages, and comes through a variety of channels. (c.) The pqysonal equation in the statement of eye-colour, when the scale contains only a list of tint-names is, I think, very considerable. The issue for the collection of data of a plate of eye-colours like that of Bertrand would be helpful, but we can hardly hope for a collection of family eye-colours so comprehensive as Mr. Galton's to be again made for a long time to come. These causes seem to me to account for a good deal of the irregularity which appears in the numerical reduction of the results, but they are not, I hold, sufficient to largely impair the very great value of Mr. Galton's material. In tabulating the data, I have followed the scale of tints adopted by Mr. Galton, and I have used the entire material available in the cases of the grandparental, avuncular, and marital relations. I nearly exhausted the data for the parental relationships, but in these tables, which were first prepared, I stopped short at 1000 for the sake of whole numbers. I found, however, that it did not make the arithmetic Digitized by Google 104 PROFESSOR K. PEARSON AND DR. A. LEE ON sensibly shorter, and I afterwards dropped this limitation. In the case of brethren I took 1500 of each case — I daresay I could have got 2000 out of the records. As the light-eyed brethren are entered ^r^^ in Mr. Galton's MS., the First Brother in my unsymraetrical tables is always lighter-eyed than the Second Brother, hence the tables had to be rendered symmetrical by interchanging and adding rows and columns before we could reduce them. Thus the symmetrical tables have an apparent entry of 3000 pairs. Of course 1500 is the number used in finding the probable error of the correlation coeflficient. The like difficulty does not occur in the brother- sister table, where indeed the difference of mean eye -colour for the two sexes would not allow of our making the table symmetrical. A comparison of the symmetrical with unsymmetrical tables for colts-colts and fillies-fillies, will show how little need there is for rendering the tables symmetrical when pairs are taken out of any similar class and tabulated without regard to the relative magnitude of the character in the two individuals of the pair, i.e., Weatherby's record places the individuals simply in order of birth and not of darkness or lightness of coat-colour. Table VII. gives the value of the chief numerical constants deduced from the twenty- four eye-colour tables in Appendix II.* (11.) On the Mean Eye-colour having regard to Sex and Generation. — In order to test the degree of weight to be given to our conclusions, I have drawn up a table o* probable errors for four typical cases — cases by no means selected to give the lowest possible values. Further, in Table VIII. I have given the probable error in the position of the median as determined in terms of the grey, blue-green range by the modification of Mr. Sheppard's formula (see p. 95). The grey, blue-green range of eye-colour is about one-fifth of the total observed range, so that the probable error in the position of the median varies from about "4 to 1 per cent, of that range. This is not a large error, but, relative to the small variations of value with generation and .sex, it is sensible, and we must not draw too fine conclusions on the basis of single inequalities. Table VI. Table of Probable Errors in Eye-colour Data. Relations. Vx Vy 1 f '* ; ^*' ^^ 1 Mother and Son . . -0253 Maternal Grandmother and Granddaughter .! -0348 Sister and Sister . . Maternal Aunt and -0244 Nephew -0230 •0188 •0350 •0244 •0186 •0431 •0767 •0414 •0267 •0276 •0216 •0255 •0256 -0283 •0314 -0361 •0216 -0234 •0250 ^0302 1 * The theoretical formulae by aid of which these constants were determined, have been indicated in the earlier part of this memoir, and in Part VII. of the present series on Evolution. The actual work of reduction has been most laborious, but I trust that our results are free from serious error. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 105 If we examine this table we see that the error in rj amounts to fix)m '02 to '025 when we have upwards of 1000 tabulated eases, but can amount to '035 when we have as few as 700 to 750 tabulated cases. An examination of the values of rj in Table VII. shows us that most of our differences with probable errors taken on this scale are very sensible. A comparison with Table VIII. shows us that the probable error of the median is always greater than the probable error of 17, and accordingly the former, being much easier of calculation, may be taken as a maximiun limit. The probable errors of f , t.c., the ratio of cTar to o-y, are more considerable, amounting to about •04 for our longer series, and even to "077 in the case of the short series of grand- mother and granddaughter, but in this case f actually takes its maximiun value of 1'291, so that the error is under 6 per cent. ; in the longer series it is under 5 per cent. Again, we see that in most cases our differences in the ratio of variabilities are quite sensible. It must be admitted, however, that the ratio of variabilities as based on the grey blue-green range of eye-colour is not as satisfactory as that based on the bay range of coat-colour in horses. In the latter case, one-half of the horses fall into the bay range, but only about a quarter of mankind fall into the grey blue-green range of eye-colour, and, fiirther, the appreciation of eye-colour seems to me by no means so satisfactory as that of coat-colour in horses. I have tried a further series of values for the ratio of the variabilities by measuring the ranges occupied not only by the tints grey blue-green, but by the whole range of tints 3, 4, 5, and 6 of Mr. Galton's classification (see p. 67). Lastly, I have taken a third method of appreciating the relative variabilities, namely, by using the method of column and row excesses, E^ and E^, discussed in Part VII. of this series. While this method has the advantage of using all and not part of the observations to deter- mine the ratio of crx to o-y, and so naturally agrees better with the results based on the four than the one tint ranges, it suffers from the evil that these excesses can only be found by interpolation methods, which are not very satisfactory when our classes are, as in this case, so few and so disproportionate. On the whole, this investigation of relative variability is the least satisfactory part of our eye-colour inquiry, and I attribute this to two sources : — (i.) The vagueness in appreciation of eye-colour when no colour scale accompanies the directions for observation {cf. p. 103, (c) ). (ii.) A possible deviation from true normality in the factor upon which eye-colour really depends {cf. p. 80, (ii) 80). Lastly, we may note that the probable error in the correlation amounts in most cases to less than '03, rising only somewhat above this value for grandparental inheritance, where our series are somewhat short — 650 to 750 instead of 1000. Here again most of the divergences are quite sensible. Allowing accordingly for the comparative largeness of our probable errors, we shall do best to base conclusions on the general average of series ; to insist on general inequalities rather than on exact (juantitative differences, and to eni4)lia8i8e the VOL. CXCV. — A. P Digitized by Google 106 PKOFESSOR K. PEAESON AND DR. A. LEE ON I i I I ? Num- ber of cases. ^ 1000 1000 1000 1000 1500 1500 1500 kC3i-lr-lfi-lt*COai COOOt^OO'^i-llOCO t*coi:*COt*l>-l>-t* OC^'*C000'*'*»0 c^i-ic<i»-ii-»i-ii-io Coefficients of regression. i 0> CO »o w »0 00 ''i^ o I-H 00 l-H o kp CO kO O Oi CO lO CO CO to ii -rt< o 1— I 00 cqf-<ioai'*f-<(Mco t^COt^CO^OOCMCO i-HcoOO'-icOi-*'^ '^cO'^cococqcNC^i loococqaoooio*^ C000C0'*'^0»C<lO COOOCOOOrHt^COCO opoiioOf7ieo»7<wo!i pI O Oi O CM t^ I-H I— 1 Oi 00 0» lO i-l ip ^ ^ ip O) CO CO CO CO i-l i-i x*4 ©q »p '^ ^ <M p ^00i-iC^t*<NO»lO <M0iC00055O0Sr-l -j*<cpcoc<i©^c<i©^T** 'Tj^OOt^Oit^t^OOlO C^Ot^COiOOC^Ii-H OOOOt-OlOCMlOOS ©^(yicpcNC^c^c^fTi Coeffi- cient of corre- lation. .^ CO o t- «o O t* »— • Oi lO CO OOO »p T*i ^ ip 0% CO lO CO CO rH .-< ^ CO »p ^ ^ (M 8 I— t CO(Mt*OiC^»OCOO r-lOi-ICOC^OC<IOO G<|00l>-Oit*G<l»Oi-» ^cpcpo^«(NC<icp '^^cogot*c<icoi-H O'^OWOOOt^O co(Ncp»7<w»7i©^©^ .2 2 »0 lO o o t^ 00 oo r-< CO 00 «o 00 p op p p CO l-H COOiCOCOO»QC<ICO 0(MCOCO^Of-«^ OiCO<3iCO»000"T**|>- ppppr^^ppb^ l-H f— 1 l-H I— 1 US000^COCOt*(N00 ^C<|i-HOOi<MOiCO eOi-ioOkAOOi-40d 1 10667 M256 •9363 1-0188 i-H I— 1 I— 1 p 00^(M»OOiO>t*Oi OSOOC<H:*»0>0000 OCOi-HCOCO<MOOOi PpppoOp»7HC<l ^ ^ ' ' ' ^ ^ lO'^G^lOCMOOiMt* i-Ht*0>(Mi-<|>-i-ii-< OOOOi-HiOOOOOi-H (ncooococooiOic^ OcO<Ni-HCOCOCOOO T»<Ol>-00iOiO00O» ppt-oppb^pt- Division of grey blue- green range by mean. ^ C<l (M t* t* <M (M CO CM O) CO o> i>> tp !>. kp l>- 00 t* r-< t* (M O CO 00 O popqp CO IO^O»(MCOCOO»0 COCOC<4lOCOkOkOO) i-Hr-<»-HCOt^i— ii—iO t-qpt-qpcpqpt^p j^ 00 ^ lo lo i-H CO 00 o» ^oo^ »p »p 00 op 00 1:* CO t* <N '^ CO 00 CO CO 00 CO co 1 lO00"^kOrHCOi-HCO COOilOr-lOiOt^t- COCOrHi-<'*i*'^C0CO '^'^(MCOOOOiOil:* t*t*oco(Mooe>o 0000kOl£d0000l>>rH COOOOlOOiOir-<<N 1 •s ^ _ ^ _ Son Daughter . . . Son Daughter . . . Brother . . . Sister . . . Sister . . . i Grandson . . Granddaughter Grandson . . Granddaughter Grandson . . Granddaughter Grandson . . Granddaughter (^ Father Father Mother Mother Brother Sister Brother 1 Paternal Grandfather . Paternal Grandfather Maternal Grandfather Maternal Grandfather . Paternal Grandmother . Paternal Grandmother . Maternal Grandmother . Maternal Grandmother . Paternal Uncle . . . Paternal Uncle . . . Maternal Uncle . . . ^latemal Uncle . . . Paternal Aunt . . . Paternal Aimt . . . 1 Maternal Aunt . . , Maternal Aunt . . . 1 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 107 I »-i -^ lO «o t:* 00 t* 50 CO CO CO CO p p p o CO c<i i-H cq Oi CO »o t* CO ^ CO CO oooo e> o 00 0> i-H Oi i-H <M <M OOO O) O CO i-H <M C^ OOO i-H:*i-<t:*'**4O00t:* 00C0»-ii-iOC0OO CO ^^ ^^ tJ^ '^^ ^* ^i^ ^* »OCO<0'^000«0»0 OOkOCOOlOC^OOG^I CO "^ CO '^ CO CO "^ kO oooooooo '^f-i»OOCq«Ot:*e> (ncoos<Mco»-ii-ii-i CO CO Cq CO CO CO CO CO «0tr*r-<«00»Oi^»-H oo(NOcoc^u:di-HO> Cs|COC0CO<MCOCO<M OOOOOOOO o 1 I I I I o O 00 '^ o I-H t* O O I-H I— I 0> lO lO «o "* <o '^ t:* «0 CO O pH ^ I-H Oi )A CO lO OOCOQOCOCOOi-HkA oocqco-^ooio Olr-iOOOOOCOOOOS OO-^COOOOSt^i-HCO coocoooqost^t* (MCOb-G<IO>OCOOO kaooioooioootct^ «* O W CO fh »0 r-« lO <0 o>^ CO o> '^i^ CO CO »o '* t* C?l CO o o» CO I-H '^ k£d CO CO C<lO>-HO)COlOO>t« CfiCOC<IOi>-Hi-H|>."T** Oit:*C<li-HOCOCO-^ cococo^co»oco^ COCOOCOi-H-^t.©^ co»«(MO>i-Ht*eoo coc<icoeocoo>coc<i -^^lO-^CO^t:*!^ CO CO (M CO Cfi CO CO I-H rH Oi I— I CO t« CO l>- t- o» t* CO »0 -^ I-H t" CO CO t^00t:*O»-H00t:*iO i-H,-<OC0CfiC0 01'* O-^^t^cocoioco lOOOOCOt^'^i-Ht* COO>COlOOO>l:«CO C^JOi-HOOCOOOOOi-H t^l:*OOI>-t*t:^l>-00 CO 0)l:« 00 T*4 00 r-Ht:^ t:^ f-H CO i-H CO CO t- t:* <3i t- O »0 -^ i-H t:^ CO O t- t- 00 O^OOCMOO^O^ CO'*<MOiOOC<|CO<M 0)i-Hi-hC^000)C0O t:^b«00t:*00t-COiO C^lOt^QOOOCOC^kA -^Ot^COOCStCMtM cqcooi-^cNi-HOo 00l>-l>-t-00l>-0000 b* e> lo »-H CO i-H OS -^ OS o ^ o »o O OS O OS t«tr*"^»-HT*<»OI>.|-H 00t*^00cOt:*O00 -^j^Qcoe^t^oooco OOOOSOSOSOSOO IOI:«COOSCOC<ICOO C<ICq>-H'^*<COt:*f-H<M 00>00000'^t:*CO osoooosooooos b" P«4 C4 OS ^ 00 CO kO G^ OS lO OS ^ OS O OS o I OS OS COC0t:*iOC5lOSt:*OS COCNCOt^'^i— lOSi-H lOOSCOt*C<lO»-H»0 OsOSOsOOf-*Oi-H OSOO'^^'^'^O^Q t^G^QOkOCOOCOCO i-Ht:^OSi-HOS00COt:^ 0»-hOSOOSi-hOsO 1 Pm •J 'J I I I I §|§|§||| 'w "^ 'o n3 "w ns t3 ts 1*1 I I ' &<S &8 9^S ^8 ^ ^ o S 2.12 n p 2 Digitized by Google 108 PROFESSOR K. PEARSON AND DR. A. LEE ON general tendency of a series rather than pick out single differences for special consideration. If we do this we shall still find that remarkable results flow from our Tables VII. and VIII., most of which seem hitherto to have escaped attention. I return now to the special topic of the present section, the mean eye-colour, after this lengthy — if needful — digression on the probable error of the data given in our tables. We may, I think, safely draw the following conclusions : — (a.) Man has a mean eye-colour very substantially lighter than that of woman. If we compare the mean eye-colour of father with mother, of son with daughter, of brother with sister, of grandfather with mother, of uncle with aimt, of grandson with granddaughter, of nephew with niece, we have the same result — man is distinctly lighter eyed than woman. (6.) Tlbcre appears to be a secular change taking place in eye-colour^ but this 15 more marked and definite in the man than in the woman. Thus we have the following mean values for 77 in classes, which must roughly represent successive generations : — Grandfather. . . -3658 1 .^aac. Grandmother . . '8757 1 .ggoo 3658 -1 .4,49 5241 J Grandmother . Mother . . . . -8757 •8290 5^29 \ -6484 7039 J Daughter. . . Granddaughter. . -7524 •8508 }^ Father }• ^"^ •^•'"•' )^ -6484 -— 8— — . . . '—J. .8016 Grandson. . . , Another comparison may be made by noting that grandsons are darker than grandfathers, sons than fathers, nephews than uncles. Similarly, granddaughters are lighter than grandmothers, daughters than mothers, but nieces are not lighter than aunts, as we might have expected. Thus, while the records show a definite darkening of the eyes of men, there appears to be a certain but less sensible lightening of the eyes of women. Again, the younger generations are much closer in eye-colour than the older generations. The difierence in eye-colour between grandsons and grand- daughters, sons and daughters, nephews and nieces is only about 15 per cent, of the grey blue-green range, but for fathers and mothers it is 30 per cent., and for grand- fathers and grandmothers 50 per cent. When we realise that difference in eye-colour appears to be a sexual character, the true explanation of this secular change in eye-colour becomes stiU more obscure. If the lighter eye-colour of the older generation be due to an effect of old age, why is it conspicuous only in the male and not in the female ? Why is the mother sensibly darker than the daughter, but the father sensibly lighter than the son ? Further, supposing light eyes much commoner among our grandfathers than among their grandsons, and dark eyes among our grandmothers than among their grand- daughters, we cannot attribute the great approach in eye-colour to a blending of the parental characters, for, as we shall see later, eye-colour does not seem to blend, it is rather an exclusive character. We should, therefore, be thrown back on prepotency Digitized by VnOOQ iC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 109 of the mother — a conclusion possibly warranted by our results in the case of daughters, but not in the case of sons. Again, why was there such a marked difference in eye-colour between the men and women of three or four generations back ?* And if it was a sexual character, why is it disappearing ? Was it not, perhaps, a racial difference ? Light and dark eyes are not unusually associated with distinct races, and it is just possible that the change in eye-colour is a product of reproductive selection ; the old blue-eyed element of the population may be dwindling owing to the greater fertility of the women of dark-eyed race, and thus without any obvious struggle for existence and survival of the fitter, the blue-eyed race may be disappearing from England, as the Langobard type has so largely gone from Italy and the Frank from France, t It will not do to be dogmatic about these matters, but the more one measures characters in different generations, the less stable do races appear to be. We speak of the national characters of the Englishman or the Frenchman based upon our experience of how these races have acted in past history, but although there has been no great racial invasion nor struggle, can we safely assert the physical characters of the Englishman to-day do not differ substantially from those of the Englishman of the Commonwealth ? It seems to me that the possibly continuous change of characters in a race, not subjected to very apparent internal or external struggle, is a problem of the highest interest to the anthropologist and .ultimately to the statesman. Whatever be the explanation of this secular change in eye-colour, it appears to correspond singularly^ enough to the secular change we have noted in the coat-colour of thoroughbred horses — in the older generation the sexes differ more widely than in the yoimger. (e.) TJie maternal male relative {grandfather^ and uncle) is substantially lighter-eyed than the patei^nal male relative {grandfather and uncle). — I see no explanation for this curious result, but it seems worth while to specially note it, for there are curious anomalies in the inheritance through the various male and female lines which may find their complete explanation some day when more and possibly more trustworthy characters have been investigated. (12.) On the Variability of Eye-colour with reference to Sex and Class. — The determination of the relative variability of not exactly measurable characters is, as we have already seen (p. 105), a somewhat delicate problem. It is more so in the case of eye-colour in man than of coat-colour in horses, for there is greater difference in the means, and accordingly the ratio of crx/o-y, as found from the ratio of the " excesses" (p. 105), will be even less reliable.^ The class indices corresponding to the * Mr. JJalton's records went back to great-grandfathers, many of whom accordingly appear in our data for grandfathers. t See Note II. at the end of this paper. J The relative variability of all classes was worked out for thorough-bred horses by the " excess " method, and in only one case — that of dam and colt — did it differ from the bay range method in its determination of the class with the greater variability. Digitized by Google 110 PROFESSOK K. PEARSON AND DR. A. LEE ON grey blue-green range are also not entirely satisfactory in their results, nor those taken for a still larger range covering tints 3, 4, 5, and 6, or blue-green, grey, hazel, light brown, and brown, which cover roughly about 1*5 to 1'6 times the standard deviation. We shall now consider the results of three methods of considering the relative variability, (a) jfrom the excesses given in columns 1 and 2 of Table VIII. ; (fi) from the grey blue -green range given in columns 3 and 4 of Table VIII. ; and (y) from the range of tints 3 to 6 inclusive given in columns 5 and 6 of Table VIII. As we have already indicated, these methods are not likely to give the same relative magnitude numerically for the variabilities ; we must content ourselves if they agree in making the ratio of cr^ to Cy greater or less than unity. Now, in the twenty-two cases a and fi disagree in 10 cases. 15 and y disagree in 7 cases. a and y disagree in 5 cases. Further, for the five cases in which a and y disagree, those for father and son, paternal grandfather and grandson, maternal uncle and nephew, show so little difference of variability in the two sexes that both methods give sensibly the same results, i.e., equality of variability. In the cases of the paternal grandfather and grandchildren, the two methods diverge rather markedly. It will be of interest accordingly to work out the probable errors as given by the excess method for one, say the first of these cases. The theory is given in Part VIL of the present series. Here Ei = 275*165, E^ = 309"013, whence we find probable error of El = 17*273, probable error of Eg = 16*925, correlation between errors in E^ and Eg = — -4424, probable error in crj/crg = '0394. Thus the probable error in the ratio of the variabilities is about 4 per cent., and of the order of the quantities by which we are distinguishing the relative size of Ci and (Tg. Further, there is another source of error in evaluating Ei and Eg due to the method of interpolation used, and this would still further increase the probable error in cTi/cTg. We cannot therefore lay any very great stress on the manner in which the ratios of variabilities for the paternal grandfather and grandchildren have swung round from (a) to (y). A further examination shows us that in all five cases wherein y differs from a it is in accord with fi. I shall accordingly take y as the standard criterion, but in those cases in which it has agreement with a, its conclusions must be given greater weight. (a.) On the Relative Variability of Sex in Eye-colour. — The following male groups are more variable than the corresponding female groups : — Sons of fathers than daughters of fathers. Sons of mothers than daughters of mothers. Brothers of brothers than sisters of sisters. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. Ill Grandsons (in four series) than granddaughters (in four same series). Nephews (in four series) than nieces (in four same series). Fathers (in two series) than mothers (in two series). Grandfathers (in foiu* series) than grandmothers (in four like series). Uncles (in four series) than aunts (in four like series). The following female groups are more variable than the corresponding male groups : — Sisters of brothers than brothers of sisters. Wives than husbands. We have thus 21 series with male preponderance against only two with female preponderance of variability. Again, the mean range of tints 3, 4, 5, 6 in 22 male series equals 1*5424 cr^, and in 22 female series equals 1*6740 o-y, or we have enough evidence to show that the ratio of male to female variability is about 1*08.* This greater variability of the male in eye-colour is of considerable interest. It does not appear to be a result of sexual selection, for so far as our comparatively small series weighs, husbands are less variable than wives. That mothers are, however, less variable than fathers seems to indicate that dark-eyed women are more fertilet than light-eyed, for we must bear in mind that mothers have on the average a darker eye- colour than wives. We have thus again reached the same conclusion as before, namely, that a dark-eyed element in the population with a prepotent fertility is replacing the blue-eyed element. The other female exception to the general rule of greater variability in the eye- colour of the male is that in mixed families the sisters appear to be more variable than their brothers, notwithstanding that brothers of brothers are more variable than sisters of sisters. In other words, so far as eye-coloiu* is concerned an exceptional man is more likely to have brothers than sisters, but an exceptional woman also is more likely to have brothers than sisters. The inference is not very strong, as the excess method (a) makes brothers of sisters and sisters of brothers of sensibly equal variability; it rests therefore on (fi) and (y) only. Still it deserves ftdler investigation. (6.) Let A and B be two grades of relationship, of which A refers to the older generation, and A and B refer to either sex. Then the variability of all the A's * It is worth noting that the ratio of male to female variability in the coat-colour of horses is r05 (see p. 96). In both cases the female is darker, i.e., has less of " colour " ; thus if we could take a coefficient of variation ratio instead of standard deviation ratio as the test, we should find the difference of variability less, possibly even zero. t For if mothers are to ]ye less variable than wives, their distribution must be more compressed round the mean than that of wives ; this denotes that fertility is correlated with eye-colour, and the darker eye- colour goes with the greatei fertility. [See Note II. at end of memoir, however.] Digitized by Google 112 PROFESSOR K, PEARSON AND DR. A. LEE ON who have female B's is invariably greater than the variability of all the A's who have male B's. The law appears to be universal, at least it is absolutely true for all the 10 cases to which we can apply it. Thus the father of sons is less variable than the fether of daughters, the maternal grandmother of grandsons less variable than the maternal grandmother of granddaughters, or the paternal uncle of nephews less variable than the paternal uncle of nieces. In other words, although women appear, in eye-colour, to be less variable than men, they spring from more variable stocks. This law is a remarkable one, but in face of the evidence for it, it seems difficult to doubt its validity. Should it be true for more characters in man than eye-colour,* the conclusions to be drawn from it will be somewhat far-reaching, however difficult it may be to interpret its physiological significance. (c.) On the Relative Variability of Different Generations. — We have already had occasion to refer to the general rule that the older generation will be found less variable than the younger, for it is in itself a selection, namely, of those able to survive and reproduce themselves. But this rule is obscured in the present case by several extraneoxis factors, thus : — (i.) The male is sensibly more variable than the female, consequently it is quite possible that an elder male generation should appear more variable than a younger female generation. (ii.) There appears to be a secular change in eye-colour going on. Thus while the grandparental population is a selection from the general population, the general population, at a given time, is a selection from that of an earlier period. Thus, taking means in the cases of the grandparental and avuncular relationships, we have from (y) the following results : — The father is more variable than son and than daughter. The mother is less variable than son and more than daughter. The grandfather is more variable than grandson and than granddaughter. The grandmother is less variable than grandson and more than granddaughter. The uncle is more variable than nephew and more than niece. The aunt is less variable than nephew and more than niece. In other words, the older generation is always more variable than the yoimger, except when rule (a), that the male is more variable than the female, comes in to overturn this law. If we confine ourselves to comparisons of the same sex the rule is seen to be universal. We are thus forced again to ask for an explanation of the decreasing variability of eye- colour, and can only seek it in that secular change we have several times had * Fathers of daughters are more variable in stature than fathers of sons (*Phil. Trans.,' A, vol. 187, p. 274). I propose to reinvestigate the question with regard to mothers from the material of my family measurement cards, which is far more extensive than the material I had at my disposal in 1895. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 113 occasion to refer to. Mean and standard deviation of eye-colour appear to have changed sensibly during the few generations covered by Mr. Galton's eye data. It is difficult to understand how any obsciu^ity about the recording of eye-colours could lead to anything but chaos in the numerical results. It does not seem to me possible that such results as we have reached under (a), (6), and (c), namely, greater variability in the male, greater variability in the stock of the female, and secular change in variability, can be due to any process of recording. I am forced to the conclusion that they are peculiar to the character under investigation, and are not due to the manner of taking the record or of dealing with it arithmetically. I have purposely avoided drawing attention to small diflPerences and forming any con- clusions which did not depend on whole series of groups and substantial averages. 1(13.) On the Inheritance of Eye-colour, (a.) Assortative Mating. — Before we enter on the problem of inheritance, it is as well to look at the substantial correlation obtained between the eye-colour in husband and wife. When in 1895 I reached the value '0931 ± '0473 for stature, I wrote, " we are justified in con- sidering that there is a definite amount of assortative mating with regard to height going on in the middle classes."* Since then we have worked out the coefficients of correlation in stature, forearm, and span for 1000 husbands and wives (instead of 200) fi-om my family datat cards. The results, which are very substantial, will be dealt with in another paper, and amply confirm my view that assortative mating is very real in the case of mankind. The value ('0931) cited above is in close agree- ment with the result now reached ('1002 db '0378) for eye-colour in the same materiaL The correlation between husband and wife for two very divergent characters is thus shown to be about "1, or is 25 per cent, greater than is required between first c(msins\ by the law of ancestral heredity. This remarkable degree of likeness between husband and wife — ^the scientific demonstration that like seeks like — cannot be overlooked. It shows that sexual selection, at least as far as assortative mating is concerned, is a real factor of evolution, and that we must follow Darwin rather than Wallace in this matter. § * (6.) Collateral Heredity. First Degree. — I deal first with this form of heredity, as it offers least points for discussion. The values of the correlation '5169 for brothers, and '4463 for sisters and sisters are considerably less than what we have found for coat-colour in horses, but, like the value '4615 for brothers and sisters, are substantially greater than '4 to be expected from the immodified Galtonian law. They could be reached by making y greater than unity in my statement of the law of ancestral heredity. || They could also be given by the law of exclusive inheritance * * Phil. Trans.,' A, vol. 187, p. 273. t See also * Grammar of Science,' second edition, pp. 429-437. X ' Roy. Soc. Proc.,' vol. 62, p. 410. § « Roy. Soc. Proc.,' vol. 66, p. 140 ei seqS II I have considered possible explanations of thia apparently large assortative mating (i.) in both stature VOL. CXCV, — A. 9 Digitized by Google 114 PROFESSOR K. PEARSON AND DR. A. LEE ON (see p. 90) with a certain degree of prepotency in the individual pairing. As we have already noted, collateral inheritance of the first degree alone considered will not enable us to discriminate between blended and exclusive inheritance. We note that the male in collateral inheritance predominates over the female, brothers being more alike than sisters in eye-colour, and brother and sister more alike than sister and sister. The mean value for inheritance in the same sex is, however, greater than the value for inheritance between opposite sexes {cf, p. 102). (c.) Collateral Heredity. Second Degree, — A very cursory inspection of the coefiicients of correlation for the eight series of avuncular relationships shows us that it is quite impossible that the mean value should be '15 as required by the Galtonian Law. The average value of the avuncular correlation is '2650, and of the regression of nephew and niece on uncle or aunt is '2733. The probable error of the former result will not be more than '02, and of the latter something greater, as the ratio of the variabilities is open to larger error. This mean value is accord- ingly, within the limits of errors of investigation, identical with the '25 to be expected on the theory of exclusive inheritance. It is a value which appears to be quite impossible on the theory of blended inheritance even with my generalised form of the ancestral law. We may draw several other important conclusions from our table of avuncular correlations :— (i.) Nephews are more closely related to both uncles and aunts than nieces are. This is true in each individual case, whether it be judged by correlation or regression. The mean correlations for uncles and for aunts are as '3081 to '2219 respectively. (ii.) Uncles are more closely related to nephews and nieces than aunts are. This is true for three out of the four individual cases ; in the fourth case the difference is of the order of the probable error of the difference. The mean correlations ot nephews and nieces are as '2923 and "2377 respectively. (iii.) Paternal uncles and aunts are more closely correlated with their nieces and nephews than maternal uncles and aunts. The mean values are as '2719 to '2580. (iv.) Resemblance between individuals of the same sex is closer than between individuals of opposite sex. The mean values for the avuncular correlation in the same sex and in the opposite sex are respectively "2751 and •2549. (v.) Uncles are more closely related to nephews than aunts to nieces (mean correlations as '3455 to '2046). In fact, generally, we see a very considerable preponderance of heredity in the male line so far as these avuncular relations for and eye-colour, being characters of local races, or even families, and the husband seeking his wife in his own locality or kin; (ii.) in a possible coiTelation of homogamy and fertility. See *Eoy. Soc. Proc,,* vol. 66, p. 28. Neither seem very satisfactory. Consciously or unconsciously, man and woman appear to select their own type in eye-colour and stature, until they are apparently more alike than such close blood relations as first cousins ! Until we know how far this correlation extends to other characters, it would, perhaps, be idle tp draw conclusions as to its bearing on widely current views as to first cousin marriage». Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 115 eye-colour extend. It is noteworthy that while the two highest correlations are reached for nephew with paternal and with maternal uncles, nearly the two lowest are found for niece with paternal and with maternal aimts. Without laying special stress on each small diflTerence, it must be admitted that the avuncular correlations vary in a remarkable manner with sex, and differ very widely from the practical equality of resemblance which we might d prioii have expected to exist in this relationship. (d.) Direct Heredity. First Degree. — Here we have a mean value of the paternal correlation = '4947. This is in excellent agreement with the '5 to be expected by our theory of exclusive inheritance ; it is thus in practical agreement with the value of the parental correlation obtained for the inheritance of coat-colour in horses. It would not be inconsistent with a high value for y in the theory of blended inheri- tance, but such a value of y is rendered impossible by the values we have obtained for collateral heredity (see 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq.). We may draw the following special conclusions: — (i.) The son inherits more strongly from his parents than the daughter, the mean correlations are as '5160 to "4733 ; (ii.) The son inherits more strongly from his father than his mother, and the daughter more strongly from her mother than her father. This is part of the general principle which we have seen to hold, namely : that change of sex weakens the intensity of heredity. The correlation of father and daughter appears to be abnormally below the other three, but something of the same kind has been noted in certain stature data ; as it is, the high correlation of father and son renders the mean paternal correlation with o&pring ('4936) sensibly equal to the mean maternal correlation ('4956). (e.) Direct Heredity. Second Degree. — If we take the mean value of the eight grandparental correlations, we find it equals '3164, while the meaji value of the regression of offepring on their grandparents is '3136. These results are absolutely incompatible with the '15 required by Mr. Galton's unmodified theory, and they in fact put the theory of blended inheritance entirely out of court. At the same time, unlike the cases of parental, avuncular, and fraternal inheritance, they cannot be said to be in good agreement with the value '25 required by the theory of exclusive inheritance. We have to admit that our grandparental data are shorter series than in the other cases, and that guesses as to grandparents' eye-colour, based on memory, miniatures, &c., were more likely to be made. Further, such guesses might easily be biased by a knowledge of the eye-colour of more recent members of the family. Still a reduction from '32 to '25 is a very large reduction, and we have to remember that for long series in the case of the thoroughbred horses, with no such guessing at colour as may occur with ancestors' eyes, we found '3353 for the maternal grand- sires, a result in excellent agreement with the '3343 found for the maternal grand- fathers in the present case. Thus while the theory of exclusive inheritance without reversion suffices to describe the quantitative values we have found for the parental, g 2 Digitized by VnOOQ iC 116 PROFESSOR K. PEARSON AND DR. A. LEE ON the avuncular and the fraternal correlation in the cases of both horse and man, it is yet in both these cases unsatisfactory so far as the grandparental inheritance is con- cerned. It may be imagined that if we allowed for reversion, we might emphasise the grandparental correlation beyond the value '25 suggested by theory. But I have shown in my memoir on the " Law of Reversion," that with the parental correla- tion as high as '5, we cannot hope to have the grandparental correlation even with reversion higher than •25. (See * Roy. Soc. Proc.,' vol. 66, p. 140 et seq.) Clearly the values obtained for grandparental correlation in this paper — the first I believe hitherto investigated — seem to present anomalies which oiu* theory of blended inheritance totally fails to accoimt for, and which may require some modification of our views on reversion before we can meet them on our theory of exclusive inheritance. I note the following general results deduced from our values of the grandparental correlations : — (i.) Grandsons are more closely correlated with both grandparents than grand- daughters are. This is true for three out of the four cases ; the exception, maternal grandmother, is covered by another rule (iv.). The mean correlation for grandparents and grandsons is '3294, and for grandparents and granddaughters '3039. (iL) Grandfathers are more closely correlated with grandchildren than grand- mothers are. This is true in three out of the four cases, the fourth being again subject to rule (iv.). The mean correlations for grandfathers and grandmothers are '3675 and "2658 respectively. (iii.) Paternal grandparents appear to be more closely correlated with their grandchildren than maternal grandparents, the average values of the two correlations being '3236 and '3097 respectively. (iv.) Resemblance between individuals of the same sex is closer than between individuals of the opposite sex. The mean values for the grandparental and grandchild correlation in the same and the opposite sexes are '3329 and '3004 respectively. (v.) Grandfathers are more closely related to grandsons than grandmothers to granddaughters, the mean correlations being as '3965 and '2693 respectively! It will be noted at once that these five rules are identical with those we have obtained for the avuncular correlations. So that there is small doubt that they are general rules relating to all grades of relationship for this character. It seems to me probable that the correct form of (iii.) is : Paternal grandfathers are more highly correlated with grandchildren ('4006) than maternal grandfathers ('3343), and paternal grandmothers (*2468)less highly correlated than maternal grandmothers (•2851). I have not stated the rule in this form, because it is not confirmed by the corresponding results for uncles and aunts. Paternal uncles ('3024) are more closely correlated with nephews and nieces than maternal uncles (*2722), but paternal aunts ('2414) are slightly more instead of less correlated with nephews and nieces Digitized by Google MATHEMATICAL CONTEIBUTIONS TO THE THEORY OF EVOLUTION. 117 than maternal aunts ('2338). I consider, however, that the correlation of paternal aunt and nephew ('2837) in our series is abnormally high. Now it will, I believe, be seen that the investigation of the eight avuncular and the eight grandparental relationships, here made for the first time,* enables us to draw far wider conclusions than when, as hitherto, only parental and fraternal corre- lations are dealt with. In making the subjoined general statements, however, I must emphasise the following limitations : — (a.) The rules are deduced only from data for one character in one type of life. (fi.) This character appears to be undergoing a secular change, a change very possibly due to a correlation between eye-colour and fertility in wcwnaan. Thus such a change might not unlikely differentiate the male and female influences in heredity. My conclusions, definitely true for eye-colour in man, and at the very least suggestive for investigations on other characters in other types of life, are : — (L) That the younger generation takes, as a whole, more after its male than its female ascendants and collaterals. (ii.) That the younger generation is more highly correlated with an ascendant or collateral of the same than of the opposite sex. (iii.) That the younger generation is more highly correlated with an ascendant or higher collateral reached by a line passing through one sex only than if the line changes sex. Thus correlation is greater with a paternal uncle than with a maternal uncle, or with a maternal grandmother than a paternal grandmother. (iv.) Males are more highly correlated with their ascendants and collaterals than females are. The above rules apply to the averages ; individual exceptions will be generally found to arise fi:om a conflict of rules. Thus (ii.) and (iii.) may in special cases come into conflict with (i.). When we have more data for a greater variety of characters, we shall see better the relative weight of these rules in cases where they conflict. [f.) Exclusive InheHtance in Eye-Colour. — A cursory examination of the eye- colour records shows at once how rare is a blend of the parental tints. Even when such is recorded, it is by no means clear that we have not to deal with a medium tint which is really a case of reversion to a medium tinted ancestor. The failure of eye- colour to blend is, I think, well illustrated by what Mr. Galton has termed cases of " particulate " inheritance. In the thousands of eye-colours I have been through, I noticed some half-dozen cases only in which the two eyes of the same individual were of different tint, or the iris of one pupil had streaks of a second tint upon it.t * I anticipate equally valuable results when characters are first correlated for the nine possible cousin series. t In the same manner the occurrence of particulate inheritance in coat-colour in horses may be really an argument against the existence of blends. In the many volumes of the studbooks I have examined, the recorded instances of pieljalds are vanishingly few in niunber. Digitized by Google 118 PROFESSOR K. PEARSON AND DR A. LEE ON If we allow that it is from the theory of exclusive inheritance that we must seek results in the present cases, we see that for parental, collateral, and avuncular relation- ships we get quite excellent results, but that the grandparental relationship is some- what anomalous. A priori it might appear that reversion would aid us in increasing the correlation between offspring and remote ascendants. But, as I have shown else- where,* this superficial view of reversion forgets that the parents as well as the offspring revert, and if we increase the grandparental correlation above '25, we at once reach difficulties in the values of the parental correlation, provided we adopt what appear to be reasonable assumptions as to reversion being a continuous and decreasing factor from stage to stage of ancestry. I am inclined accordingly to suspend judg- ment on the grandparental relationships, thinking that the smallness of the number of families dealt with in Mr. Galton's data (200) may have something to do with my peculiar results. Meanwhile I shall endeavour to get the remaining six grandparental tables for thoroughbred horses worked out, and see whether they confirm the high values ah'eady found for the two maternal grandsires and oflfepring, or give an average value much nearer '25. That the reader may see at a glance the general results hitherto obtained in this and other papers, I append the following table of inheritance : — * See my paper on "The Law of Reversion/' 'Roy. Soc. Proc./ vol. 66, p. 140 et scq. Also *The Grammar of Science/ second edition, 1900, pp. 486-96, " On Exclusive Inheritance." Digitized by VnOOQ iC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 119 Table IX. — Theoretical and Actual Results for Inheritance. Relationship. Theory. Man. Horse. Hound. Daphnia. Blended inherit- ance.' Exclusive inherit- ance.* Statiu^e*. Head index.* Eye- colour.* Goat- colour.^ Coat- colour." Spine." Parental .... Mid-parental . . . Grandparental . . G. Grandparental . Avuncular . . . Whole sibling . . Half sibling . . . •3000 •4242 •1500 •0750 •1500 •4000 •2000 •5000 •2500 •1250 •2500 •4 to 10 •2 to -5 •3355 •4745 •4034 •3348 •4735 •4025 •4947 •3166 •2650 •4749 •5216 •3353 •6329 •3550 •3507 •1340 •0404 •5170 •1646 [•3295] •4660 [•1360] •6934 * Mr. Galton's unmodified hypothesis. See " Law of Ancestral Heredity," * Roy. Soc. Proc./ vol. 62 p. 397. 2 Without any reversion. See "Law of Reversion," * Roy. Soc. Proc.,' vol. 66, p. 140 et seq. The values for the fraternal correlation depend on the degree of prepotency of either parent within the union. 8 See * Phil. Trans.,' A, vol, 187, p. 270. ^ See 'Roy. Soc. Proc.,' vol. 62, p. 413. The paternal correlations, for reasons stated in the paper, are excluded from the result. ^ See p. lis et seq. of the present memoir. ^ See p. 98 et seq. of the present memoir. The grandparental correlation is based on two cases only. 7 See *Roy. Soc. Proc.,' vol. 66, p. 140 et seq. ^ Sec * Roy. Soc. Proc.,* vol. 65, p. 154. I have deduced the value for parents and grandparents from Dr. Warren's results for midparent and midgrandparent. The value for whole siblings I obtained from Dr. Warren's measurements, which he with great kindness placed at my disposal. (14.) Conclusions. — The course of this investigation has not been without diffi- culties, and I am fully prepared to admit that more obscurity and greater probable errors are likely to arise when we deal with the inheritance of a character not directly measurable, than when we take that of a character to which we can at once apply a quantitative scale. But I contend that many of the characters, the inheritance of which it is most important to investigate, do not at present, and perhaps never will, admit of a quantitative measurement. We can arrange in order, we can classify, we can say more or less intense, but we cannot read off value on a scale. It is just such characters also, which the not highly trained observer can most easily appreciate and record. Hence we have been compelled to devise some method of dealing with them, and the present paper illustrates how the method invented can be applied to reach results of considerable interest and of substantial validity. Digitized by Google 120 PROFESSOR K. PEARSON AND DR. A. LEE ON In order to illustrate the method, I chose two characters, coat-colour in horses and eye-colour in man, which seemed sufficiently diverse both as to origin and species.* The new method enabled me to reach results for half-brethren, grandparents and uncles and aunts, which had not yet been independently considered. The conclu- sions arrived at for eye-colour in man at no point conflict with those for coat-colour in horses, and both in the main accord with the theory of exclusive inheritance with- out reversion herein developed. We find — (i.) No approach to a single value for the coefficient of inheritance for each grade of relationship; it varies widely with the sex, and the line through which the relationship is traced. (ii.) No approach in average values to those which would be indicated by Mr. Galton's Law. Nor does the modification of Mr. Galton's Law, which I have termed the Law of Ancestral Heredity, give better results. For, if we cause it to give the parental values, it then renders results inconsistent with the fraternal values. (iii.) There is agreement with the theory of exclusive inheritance without reversion for the parental, avuncular and fraternal series ; but there is some anomaly in the case of grandparental inheritance. This requires further investigation, and possibly a modification of our views on the nature of reversion. We want a list formed of characters in various types of life, which are supposed to be exclusively inherited, and then experiments ought to be made and statistics col- lected with regard to these characters. It is in this field of exclusive inheritance that we must look for real light on the problem of reversion. If we consider the three known forms of inheritance, the blended, the exclusive, and the particulate (which may possibly be combined in one individual, if we deal with different organs) ; if we consider further that these forms may possibly have to be supplemented by others not yet recognised {e.g.y reversional theories depending, say, on heterogamous unions), then it would appear that the time is hardly ripe even for provisional mechanical theories of heredity. What we require to know first is, the class of organs and the types of life for which one or other form of inheritance predominates. As variation in no wise depends on the existence of two germ-plasms, so biparental heredity can by no means be treated as the result of their simple quanti- tative mixture ; the component parts of these germ-plasms corresponding to special characters and organs, must be able to act upon each other in a variety of qualita- tively different ways. To adopt for a moment the language of Darwin's theory of pangenesis, the multiplying gemmules from an organ in the father must (i.) cross with gemmules from that organ in the mother, and the hybrid gemmules give rise to blended inheritance, (ii.) must without crossing multiply alongside the gemmules of the mother, and give rise to particulate inheritance, (iii.) must alone survive, or alone * Since supplemented by my investigations, based on Mr. Galton's data, for coat-colour in hpunds, * Roy. Soc. Proc.,' vol. 66, p. 140 et seq. Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 121 be destroyed in a struggle for existence with those of the mother, and give rise to exclusive inheritance. And all these three processes may be going on within the same germ-plasm mixture at the same time ! Even without using the language of gemmules, processes analogous to the above must be supposed to take place. Thus a quantitative " mixture of germ-plasms " becomes a mere name, screening a whole range of mechanical processes ; and very possibly a new one could be found for each new form of heredity as it occurs. Such processes like the old ones would still remain without demonstrable reality under the veil of " mixture of germ-plasms." What I venture to think we require at present is not a hypothetical plasmic mechanics, but careful classifications of inheritance for the several grades of rela- tionship, for a great variety of characters, and for many types of life. This will require not only the formation of records and extensive breeding experiments, but ultimately statistics and most laborious arithmetic. Till we know what class o^ characters blend, and what class of characters is mutually exclusive, we have not within our cognizance the veriest outlines of the phenomena which the inventors of plasmic mechanisms are in such haste to account for. Such inventors are like planetary theorists rushing to prescribe a law of attraction for planets, the very orbital forms of which they have not first ascertained and described. Without the observations of Tycho Bbahe, followed by the arithmetic of Kepler, no Newton had been possible. The numerical laws for the intensity of inheritance must first be discovered from wide observation before plasmic mechanics can be anything but the purest hypothetical speculation. Appendix I. Tables of Colour Inheritance in Thoroughbred Racehorses, extracted by Mr. Leslie Bramley-Moore jfrom Weatherby's Studbooks. Table of Colours. 1 = black (bl.) 9 = chestnut or bay (cli./b.). 2 = black or brown (bl./br.). 10 = chestnut (ch.). 3 = brown or black (br./bL). 11 = chestnut or roan (ch./ro.). 4 = brown (br.). 12 = roan or chestnut (ro./ch.). 5 = brown or bay (br./b.). 13 = roan (ro.). 6 = bay or brown (b./br.). 14 = roan or grey (ro./gr.). 7 = bay (b.). 15 = grey or roan (gr./ro.) 8 = bay or chestnut (b./ch.). 16 = grey (gr.). VOL. CXOV. — A. R Digitized by VnOOQ iC 122 PROFESSOR K. PEARSON AND DR. A. LEE ON Cdis. J a 1 c5 -4^ I e1 QQ t^ '^ I-H cq CO ;}: I-H O I-i r^ o o o o lO I-H & O o o I-i o r^ r^ o o CO o o o o o o «o O o o o o o o o o o o o o o o o o r-H 2 o o o o o o o o o o o o o o o o o CO rH t o o o o o o o o o o o o o o o o o ■4 o o o o o o o o o o o o o o o o o • i-i 4 o o o rH o o o o o o o o o o o o r^ d -i t^ o o * o 00 cq cq l-H o o o o o o o oq cq «o CO o> 4 o o o o o o o o o o o o o o o o o o 00 o o o o o o o o o o o o o o o o 1?^ ^ ^ eo § l-H CO CO oq 00 CO o l-H 5: r^ o o o o CO I-H «o «o ^ ^ o r^ o -**< o o I-i o o o o o o o o o o o o o o o o o o o o o o o o o o o •«* i t* cq r^ CO o cq 00 o o 55 o o o o o o i CO ^ i o o o o o o r^ o o o o o o o o o • 3 o o o o o o o o oq o o o o o o r^ 3 CO o o cq o o o o r^ o o o o o o t* 3 1. 3 i ^ i ^ 1 ^ 4 J. 4 s s 2 ^ & ^ • I^ c^ CO ''I* lO to Jb-» 00 o> o I-I cq 00 ^ r^ I-i CO • 1 Digitized by VjOOQ IC MATHEMATICAL CONTEIBUTIONS TO THE THEORY OF EVOLUTION. 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LEE ON CoUb. i 1 J 1^ •rj tJ I J CO rH *- I-H 5 r-H lO CO s o o i o o o o t—i 01 r^ CO & o o o o o o ^ o o CO o o o o o o t^ r-H o o o o o o o o o o o o o o o o o rH o o o o o o o o o o o o o o o o o I-H s o o o o o o I-H o o <N o o o o ^H o -* r-H o o o o o o r^ o o o o o o o o o rH r-l o o o o o o o o o o o o o o o 1^ r-i o I-H 4 '^ o o 00 f-H I-H CO CO 00 o o eo CO I-H o o o o o r-i 5 o> o o o o o o t—i o o o o o o o o o I-H 1 00 t ^ o o o o o o o o o o o o o o o o o ^ ^ eo ^ o o CO I-H I-H o o r-H o o o o o o <o 1. o o o lO o CO CO o o cq o o o o o o CO r-H »o e ^ o o o o o o o o o o o o o o o o o '^ i eo cq o eo I-H CO o o t- C9 o o o o o o o o CO ^ i o o I-H o o o o o o o ^ o o o o t-* C<I 4 3 I-H o o r^ o o I-H o o o o o o o o o eo l-l 3 lO rH o CO o r^ o> o o CO 4 o 2 o o g o o o 00 :i 1- 3 M e i ^ e 4 & ? 4 bo ^ : r^ <N CO ^ iO CO t* 00 a> o r-H r-H I-H eo I-H I-H CO I-H i Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 125 FiUies. J I i 8 p 'g 3 S a o o lO I-H CO o «o ^ ^-1 o a> o o rH rH Ci| t* i-« ^-c ''I* lO <o a> o r^ -**< cs o H • r-l & o o o o o o rH o o CO o o o o o CO t* to 2 o o o o o o o o o o o o o o o o o I-H & ^ i> o o o o o o o o o o o o o o o o o r^ S CO I-H 2 o o o rH o o rH o o o o o r^ I-H o o -* M <M o o o o o o o o o o ^-1 o o o o o o I-H I-H t rH 2 o o o o o o o o o o o o o o o o o r«H -g d .d c^ O^l o 00 o <o a> r^ o 55 o o o o o cq CO i-M t) I-H l>- ^-1 § o> ,0 o o o o o o o o o f-l o o o o o o rH " ^ 00 :« ^ o o o o o o r^ o o o o o o o o o r^ . ^ ^ 00 o t- o r-H 00 o o pi o o o o r^ r^ o. fjQ CD CO 00 C9 o I-H § <d i-H o r-l e<9 o I-M CO o o l>- o o o o o o 00 I-H id o o o o o o o o o o o o o o o o o '^ »: I-H '^ o c^ o lO ^ o o t* o o o o r^ I-H lO ^ »o rH l>- <N F^ CO i o o o o o o o o o o o o o o o o o c<i <! 3 I-H o o o o o I-H o o f-H o o o o o o CO ,^ I-H l-l o CO o CO CO o o -**< o o o o o o o 3 ^-c (M 3 3 ^ i i rd 1 2" ■i 2 s 2 -1 2 2^ & : I-H (M CO -**< lO o t^ 00 o> o I-H I-H I-H CO r-H '* ^-1 CO H Digitized by VjOOQ IC 126 PROFESSOR K. PEARSON AND DR. A. LEE ON CoUs. i I 1 3 ^ i i I •I I -« i n e3 1 00 CO '^i^ CD CO 2g tr o o •^ o o f-< I-H o lO i-i r^ I-H ^ s § f-H 1-4 & o o o I-H o o pH o o I-H o o o o o- r-H -**< d o u o o o o o o O o o o o o o o • o o o r-H fe o '^ .& o o o o o o o o o o o o o o o o rH 2 CO 2 o o o o o o o o o o o o o o o o o ,4 (M 4j^ o o o o o o o o o o o o o o o o o rH 2 I-H 2 o o o o o o o o o (M o o o o o o <M r-H 4 o 4 -«»« o r-H a> o C<l o o o 00 o o o o o I-H to r^ G^ I-H o (M t* ^H r-H o <N O) 4 o o o o o o o o o o o o o o o o f t) ] 00 o o o o o o o o o o o o o o o o o ^ (M I-H C<l -* f-H '^i^ «o o o CO o o 1-H r^ o <N ^ t^ rfi r-H I-i Oi (M 00 (M I-H 2 <o 4 ^ I-H o o C<l o CO 00 o o I-H o o o o o I-H to I-H to o r-H o o o o o o o o o o o o o o o o gl C<1 I-H a> (M t^ r-H o o t* o o o o o o o il (M kO I-H I-H I-H ^ CO o o o O o o o o o o o o o o o o o (M i 3 o o o o o o I-H o o C<I o o o o o o CO I-H 3 o o o I-H o (M o o o o o o o o o o CO 3 4 3 i 4 ^ e ^ ^ 4 o g^ ^ 2 1^ ^ : i-H « CO '^ \Q CD t* 00 a r-H (M CO -^ »o (D i ""^ *"* o H Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 127 Fillies. •I i 1 I 4 o O I I 1 1 3 ■i! o Oi CO r-H r^ 00 o l-H o t^ r^ o C<l o r-H CO ^ S CO CO -* t^ O) o I-H ^ <N ^ H r-i CO rH & r-H o o o o o o o o r-H o o o o o (N ^ d to o o o o o o o o o o o o o o o o o ^ ^ ^ o o o o o o o o o o o o o o o ' o o g CO 2 o o o o o o o o o o o o o o o o o ^ ^-1 o o o o o o o o o o o o o o o o o I-H -2 o o o o o o o o o I-H o o o o o o r-H -g d ,d 00 T-t (M I-H o t* I-H o o CO o o r^ o o C<l o> f-H o CO o I-H •-H CO O^ e o o o o o o o o o o o o o o o o o t? ■€ ^ ^' g 00 ;«_ o o o o o o o o o o o o o o o o o cs ^ ^<« ^ CO CO r^ (M r^ r-^ lO r-l o r^ r-H o rH o r^ cq Oi r-H l>- CO s r-H g <p 4 ^ o o o '^ o CO kO o o CO o o o o o o ko 4 o o o o o o o o o o o o o o o o o ^ i lO 01 o C<4 o CO s o o s o o o o o o Ol • CO 5. o o o o o o o o o o o o o o o o o fj « 3 o o o C9 o r-H CO o o I-i o o o o o o t* f-H 3 o o o o o o l-l o o o o o o o o o r^ 3 u e s ^ i rS ^ ^ ^ 4 g 3" 4 s i. ^ & : rH c^ CO -**< lO CO t* 00 pi o I-H C4 CO ^ lO CO • ""* "^ "^ '^ *"* "^ H Digitized by VjOOQ IC 128 PROFESSOR K. PEARSON AND DR. A. LEE ON Second CoU. s (D M ■^ o u W ^ CO 3 £ w CO ^ a d 2i So sS ^ T Q ■+2 .^ 3 s rO pC4 S I (D •s OQ 1 .^ '1 a © •4^ Ti ^ § <? ,^ 1, a g H a 1 CO eo c^ 00 r-^ CI -^ o I-H CD o o C9 o o 00 ^ (M r-H CD I-H ^ !§ 00 § H r^ r-H ^ o o o o o o I-H o o o o o o o o C9 CO id 2 o o o I-H o o o o o o o o o o o o I-H 1— 1 & I-H 2 o o o o o o o o o o o o o o o o o eo o o o o o o o o o o o o o o o o o o T-H h — M I-H o o o o o o o o o o o o o o o o o d ^-C (1 o o o o o o o o o o o o o o o o o l-l 4 d rd lA CD o ^ I-H GO O) o o '^ o o r*i o o eo I-H rH cS o O o CO r-H I-H oo I-H CO O^ e o o o o o o o o o o o o o o , ^ ^ o o J "? ^ 00 !y o o o o o o o o o o o o o o o o o ^ t^ ^ 00 >o r-H (N o O) GS o r^ O) o o r-i o o <M o t^ V— 4 '^ I-H t^ (M r-H ^ CO .1. rf5 1— 1 o I-H o o o I-H pi r-H o o to o o o o o o ^ kd o o o o o o r-H o o o o o o o o o I-H -*i5 ^ CD C<l o ''i* o -^ « o o ''i* o o o o o I-H CO x> "^ CD ei ^ f-H .-i eo o o o o o o cq o o o o o o o o o cq c<i 4 3 I-H o o CO o o CO o o <N o o o o o o 04 1-3 IB C<l o o '^ o r-H id o o C<l o o o o o o 1^ 3 e s. M ^ ij ^ rd :S 4 4 2^ i g ^ & : 3 i Jst rfi ^ iJ 2 2 6b i-< ^ «0 ^ ud to t^ eo 04 o I-H I-H C4 I-H CO 1-H f-H CD ^H i Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 129 I S 2 1 i W 6 i - 1 co -«*< CO c^ s o I-I o o cq o iH rH I-I i rH & •o o o ^-c o o CO o o CO o o o o o ^ r^ to 1— 1 4. o o o r^ o o o o o o o o o o o o IH o o o o o o o - o o o o o o o o o o CO I-H g o- o o o o o f-l o o I-I o o ■ o o o o « c4 1 o o o o o o o o o o o o o o "O o o l-H rH 4 o o o o o o o o o o o o o- cr- "O o o -i t^ 00 o 2 1-1 CO T-t cq o o oq o o iH o o CO s kO 1 o> o o o o o o I-H o o o o o o o o o iH ! 1 o o o o o o o o o o o o o o o o o i CO 00 CO 1-^ r-i s§ 1 o 1-^ o o I-H o o CO CD C<l o I-« o o s o o CO o o o o o o 00 00 id ■^ i o o- o - o o o 1— 1 o o rH o o o o o o cq ^ ^ o I-i iO ■ o g o I-H o o 00 o o o o iH fH IH CO CO A. 3 o o o o o o o o I-I CO o o o o o o o o o -* kO o 00 o o 00 o o o o o o • :i ^ ■l-l o o o oq eo 1— 1 o o t* o o o o o o s 3 4 3 M ^ i ^ ^ ^ 4 4 -8 g 1^ & : f-H eq CO ^ la <D t* 00 Oi o fH ' »H IH iH eo FH -«*< r--* IH IH i VOL. CXOV. — A. Digitized by VjOOQ IC 130 PROFESSOR K. PEARSON AND DR. A. LEE ON Second Colt. 1 00 -* (M 00 1-f CO 00 o o Oi o o r-l r^ o oq ^ r-H CO CO 00 o O ^H H o CO c> o o o o o o r^ o o o o o o o cq CO 1 r-< to 1 d l o o o o o o o o o o o o o o o o 1 ^ 1 ,,-^ 1— < ^ o o o o o o o o o o o o o o o o o 00 o frt t-, 4) M 1 CO 2 o o o 1-f o o o o o f^ o o o o o o <N u pq I-H t o o o o o o o o o o o o o o o o i o 1 2 § T-t 2 o o o o o o o o o o o o o o o o o I-H . 02 -s © So ^ d I-H -i o o o g o CO I-H 00 o o CO o o o o o o 00 1— 1 oq © 22 ^ i ci o o o o o o o o o o o o o o o o o ►> 00 o o o o o o o o o o o o o o o o o ."tj ^ o o o CO , ^ 00 o f-H It- o t^ o o o Oi o o r-l w 6 t^ CO l-l 00 ij . 2 g <d ^ o o o oq o oq 00 o o '^ o o o o o o CO i 1 -2 3 rJCJ id e i o o o o rH o o o o o o o o o o o r^ 1 H •— t ^* i t* CO o CO o o o o o o I-H o o §8 P> 1 CO o f-H o o o o 1— 1 o o o o o o o o o c\ •«) pO EH fi^ d 3 o o o T-t o o -**< o o c^ o o o o o o t* PH 3 CO o I-H kO o o o> o o ei o o o o o o o cq 3 t i i e i ^ ^ ^ ^ -§ 4 4_ t 1 i 2 ^ : T-l C<1 CO -* lO <o t* 00 o o rH CO r-l CO 1 Digitized by VjOOQ IC MATHEMATICAL CONTMBUTIONS TO THE THEORY OF EVOLUTION. 131 o Q P 1 ^ ■f3 06 ^3 k ^ 1 00 eo 1-4 -* to C4 C9 I—* o o 1 o o CO T-t o lo (M to 1-4 & O o o o o o f-H o o o o o o o o -^ to to ^-1 -1 o o o o o o o o o o o o o o o o o r-l 2 o o o ^-c o o o o o o o o o o o o 1-4 2 o o o 1-4 o o 1-4 o o 1-4 o o o o o o CO c4 1— 1 1 2 o o o o o o o o o o o o o o o o o ^-1 I-H ^ -i o o o o o o o o o o o o o o o o o d I-H ©^ e«i o o t^ o o 00 eo o o l-l o o o to j oS o o o o o o o o o o o o o o o o o ! 00 ^2" o o o o o o o o o o o o o o o o o 1 t-* ^' I—I -**< C<l 5 o s o o o o fH o o IH <d 1. o o o 1-i o -**< s o o t^ o o o o o o s to ^ i o o o o « o o o o o o o o o o o (M ^ i ©^ '^i^ o I-H o T-t 1—t 1— 1 o o iO o o f-H iH o o ^ ^ CO ^ i rH 1-4 o o o o cq o o o o o o o o o •«* (M* o o rH ^ o o -**< o o cq o o o o o o 1—t rH ;^ <o o f-l o o I-H o o C<l o o o o o o 00 CO 3 ^ 3 i e i ^ ^ ^ 4 ^ 4 4 4 4 2 2 2 1 & ^ • I-H <m' PS '^i^ o <o t^ 00 o> o I-H I-H eo rH IH to 1 s 2 Digitized by VjOOQ IC .132 PROFESSOR K. PEARSON AND DR. A- LEE ON Second FiUy. s .§ ^ QQ •S ■y QQ § % jS w ''S -4^ P5 t^ rO ■*s © •"O ^ g 1^ W ^ '3 y tri r/) -^ i 9 ?3 1 PR M 13 M g Gi t3 S PC H P^ 4 ^ 00 rH T-^ o o CO rH o 00 r^ o ^ o cq ^« oq ^ ^ r^ lO o rH lO C<l o EH r-i & O rH o o o o (M o o 1^ o o o o o r^ to lO 2 O O o o o o o o o rH o o o o o o r^ ^ o o o o o o o o o o o o o o o o o CO g o o o o o o o o o o o o o o o o o ^ o o o o o o o o o o o o o o o o o • r-l g o o o o o o o o o o o o o o o o o r-l -^ d rd -* CO 1^ lO o *- CO r'H o a> o o <M o o f^ CD CO rH «? c^ o (M 0!» ^ o o o o o o o o o o o o o o o o o -^ •g 00 -§ o o o o o o o o o o o o o o o o o iS" t^ ^ CO r-l CO o CD o CO 01 CO ^^ CO o o rH o C9 o r-^ cq CO «> rH o o lo o to o o o o o o o r^ o 5 id o o o o o o o o o o o o o o o o o -* i '^ rH o SI o CO CO o o CO o o o o o CO 1-^ i-i CO o o o o o o r^ o o o o o o o o o r^ M c^ 3 o o o 1^ o 1-^ CO o o lO o o o o o o o I-H 3 (M o o ^ o o o> o o o o o o o o o g 3 3 §. u Xi i 1. ^ -i ^ 4 g g g g^ ^ : r-H <N CO -t** lO CO It- 00 Oi o T-H CO '^i^ ia CO 1 '^ *"* "^ *"* Eh Digitized by VjOOQ IC MATHEMATICAL C50NTBIBUTION8 TO THE THEORY OF EVOLUTION. 133 .3 OQ 1 OQ ■4^ , £ $ ^ s° •-e if 1 I ►> § GQ 1 t5^ 1 •■2 m 3 e1 H 00 I-H C<1 I-H o s CO B 1-4 I-H o -^ s I-H o '^ o CO I-H i CO ^ O I-H o eo o o '^ o o cq o o o o o o o o o o cq 1-4 to i-H 1. O o o o o I-H I-H o o I-H o o o eo -1 o o o o o o o o o o o o o o o i-H 2 o o o o o o <n o o cq o o o o o o '^i* I-H o' o o o o o o o o o o o o o o o o 1-4 1-4 I 4 o o o o o o I-H o o o o o o o o o I-H d 1-4 4 a 00 I-H I-H to o I-H I-H I-H o o o <M o p-l <M to o» 4 o o o o o o o o o o o o o o o o o 00 t ^ o o o o o o o o o I-H o o o o o o I-H t^ ^ CO I-H 8 I-H o '^i* '^i* CO CO o o 35 I-H I-H o <M o I-H '^ CO I-H CO 1^ 1-4 o 00 o I-H '^i* ^ o o I-H o o o o o o I-H o s to ^ M o o o o o o o o o o o o o o o ^ i 00 c^ o s o 00 I-H o o p-l to o o o o o eo p-l S5 eo o o o o o o I-H o o I-H o o o o o o (N <H 3 o o o cq o I-H CO o o 00 o o o o o I-H 00 p-l rH :i '^ o o 00 o I-H o o o» o o o o o o p- 1 1. 3 M ^ ^ ^ t ^ e ^ ^ 4 4 s 2 2 & i O^l CO ^ . to CO t* 00 Oi o I-H I-H p-l p-l eo I-H p-l to I-H CO I-H i Digitized by VjOOQ IC 134 PEOFESSOE K. PEAESON AND DE. A. LEE ON Second Filly. s .a OQ fe CO ^ 00 J J 1 i 1 1 T @ 1 S 00 o CO I-H o CO CO o o o o o o o to I-H I-H ^ o o o o o o I-H o o o o o o o o '^ kO id 1-^ o o o o o o o o o o o o o o o o o h3 2 o o o o o o o o o o o o o o o o o CO 2 o o o o o o o o o o o o o o o o o o o o o o o o o o I-H o o o o o o I-H i-H p- 1 4 o o o o o o o o o o o o o o o o o d p- 1 •§ T*< o o I-H o CO o o 00 CO I-H o o o o o o CO ■« ^ a> 4 o o o o o o o o o o o o o o o o o 00 J o o o o o o o o o o o o o o o o o Oi i>^ ^ I-H CO . o CO to o. 00 I-H CO CO CO o o o o o o o I-H CO i ^ I-H <M o '^ ■ o ^ i-H o o CO o o o o o o I-H CO lO e i o o o o o o o o o o o o o o o o o g »-H -* i CO <n o CO o lO § o o CO o o o o o o CO o o o o o o o o o o o o o o o o o cq ^ 3 I-H o o kO o I-H <M o o I-H o o o o o o o I-H I-H 3 CO I-H o CO o 1-H 00 o o '^ o o o o o o CO 3 i 3 1. ^ 4 1 2 2 2^ & : I-H cq CO -^ id CO fc* 00 o> o I-H I-H I-H C9 I-H CO I-H I-H kO I-H CO 1-H i Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 135 e I .a 0) OQ 2 I I 1^ I ^ 1. 1 1 00 o 1 o eo CO o o i o r^ o o o o I-H i 1-^ &) o o o o o o cq o o o o o o o o 00 o I-H id 2_ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CO f— 1 g o o o o o o o o o o o o o o 'O o o 4 1 o o o o o o o o o I-H o o o o o o I-H I-H 1-^ 4 o o o o o o o o o o o o o o o o o o 1-^ 4 00 1-H o ^ o CO r-H o o <o CO CO o I-H o o o o o j o> ■5. 4 o o o o o o o o o o o o o o o o o o o 1*^ 00 4 o o o o o o o o o o o o o o o 1 l>^ ^ I-H lO o I-H o u CO o o to I-H o o o o o <M 6q' ^ (M CO o a o 00 CO o o CO o o o o o o CO CO id o o o o o o o o o o o o o o o o o H* i I-H t* o 00 I-H o Oi I-H o o :$ o o o o o o i eo o o o o o o o o o o o o o o o o o e<r 3 cq o o t^ o eo kO o o I-H o o o o o o 00 I-H 1—4 3 ©51 I-H c^ o I-H o C<1 I-H o o 00 o o o o o o i§ 3 3 i 4 pQ ^ i ^ 4 4 1 4 -§ 1 2 2 2 d) : I-H cq CO T*4 kO CO t^ 00 Oi -O I-H I-H I-H cq I-H CO I-H I-H »o I-H CO I-H i Digitized by VjOQQ IC 136 PROFESSOR K. PEARSON AND DR. A. LEE ON Colts. u I § i .a -g ^ I i ^ I ^ v- CO >., OB ^ I I ^ 00 lO 1^ g »o 1:* o o cq o o <M o I-H CO I-H s & o o o o o I-H cq o o o o o o o o P-H ^ 15. gr./ro. o o o r-^ o o o o o I-H o o o o o o cq "1 o o o o o o o o o o o o o o o o o 2 8 o o o 1-^ o o CO o o I-H o o o o o o kO 12. ro./ch. o o o o o o o o o o o o o o o o o 11. ch./ro. o o o o o o o o o o o o o o o o o 2-g t^ o o g? I-H kO I-H 1-^ o o I-H I-H o o I-H o o o 9. ch./b. o o o o o o o o o o o o o o o o o 8. b./ch. o o o o o o o o o o o o o o o o o »^ ^ <M o l-l o 00 CO CO o o to I-H o o I-H o f-^ f-^ 1 00 «4 o o o 00 1^ o I-H o o kO I-H o o o o o 1-H lO 4. 5. br. br./b. o o o o o o o o o o o o o o o o o ^ C<1 p-l ^ e<i CO I-H <o o o 00 cq o o o o o o I-H 3. br./bl. o o o o o o o o o o o o o o o o o 2. bl./br. o o o o p- 1 o 00 o o (M o o o o o o 1-H I-H ^ 3 o 1-^ o eo o o »o o o '^ o o o o o o CO I-H 3 3 i «5. ^ ^ ^ -i 4^ 4 2 2 1^ ^ : 1-^ o^ CO •^ lO <o t* 00 a o I-H I-H I-H CO I—I I-H lo 1— 1 <o I-H 1 Digitized by VjOOQ IC MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 137 Cdis. GQ ^1 1^ I I Q 1 ^ I I 1 § T*< p-i p-i I-H CO o o I-H o I-H o o o o o o OS I-H o o o o o o 99 o o o o o « '^ 16. gr./ro. o o o I-H o o o o o o o o o o o o I-H 14. i-o./gr. o o o o o o o o o o o o o o o o o 2 g o o o o o o o o o I-H o o o o o o I-H «4 i " 2 o o o o o o o o o o o o o o o o o 10. 11. ch. ch./ra o o o o o o o o o o o o o o o o o o o CO I—I o cq o CO I-H I-H o o o o o CI C9 C9 9. ch./b. o o o o o o o o o o o o o o o o o 8. b./ch. o o o o o o o o o o o o o o o o o *^ ^ <D I-H o CO I-" I-H I-H s ss o o s I-H o o o o (N kO 6. b./br. o o o O) o CO 09 I-H o o '^i* o o o o o C9 o CO 5. br./b. o o o o o o o o o o o o o o o o o ^ i lO o o CO CO o to CO CO o o CO C9 o o o o o o CO I-H o 3. br./bl. o o o o I-H o o o o o o o o o o o o o o 2. bl./br. o cq o »o o o I-H o o o o I-H o I-H -• 3 CO o 1^ CO o - QO o o f-^ o o o o o o s 3 1, 3 } i 4 ^ t ^ 4 1 ■i 4 1 g 2 ^ ^ & : i-H eq eo ^ lO CO t^ 00 Oi o I-H I-H I-H C9 I-H CO I-H I-H I-H CO I-H i VOL CXCV. — A. Digitized by VjOOQ IC 138 PROFESSOR K. PEARSON AND DR. A. LEE ON Appendix II. Tables of Eye-colour Inheritance in Man, extracted by Kabl Pearson from Mr. Francis Galton's Family Records. 1 = light blue. 2 = blue, dark blue. 3 = blue-green, grey. 4 = dark grey, hazel. Table op Tints. 5 = light brown. 6 = brown. 7 = dark brown. 8 = very dark brown, black. This grouping is not quite in keeping with more recent divisions of eye-colour, but being that adopted by Mr, Galton in his original collection of data, it could not be modified in accordance with present practice. Tables for the Direct Inheritance of Eye-colour. First Generation. I. — Fathers and Sons. 1000 Cases. Failiers. GQ Tint. 1. 2. 3, 4. i 5. 6. 7. 8. Totals. 1 9 12 5 5 , 1 2 34 2 10 163 65 36 ' 1 7 15 4 301 3 10 73 124 41 1 12 18 6 284 4 4 21 34 55 11 11 1 137 5 1 2 2 1 5 6 1 26 12 19 ' 1 19 16 6 100 7 1 23 16 14 , 11 31 2 98 8 1 4 8 10 1 7 10 41 Totals 36 322 264 180 ; i 5 64 101 28 1000 II. — Fathers and Daughters. 1000 Cases. Faihers, Tint 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 4 9 11 4 1 2 1 32 'n 2 11 139 57 31 6 24 6 273 ^ 3 9 73 111 38 1 15 19 3 269 1 4 5 43 34 54 2 10 14 3 165 5 1 3 3 1 8 M 6 1 45 13 19 23 15 3 11? 7 2 27 10 12 7 41 6 105 8 8 4 2 11 4 29 Totals 32 345 243 158 3 67 127 25 1000 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEOEY OP EVOLUTION. 139 in. — Mothers and Sona 1000 Casea Mothers. Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 5 14 6 3 1 6 35 2 12 119 83 29 8 20 21 9 301 3 13 54 113 35 4 37 14 8 278 4 3 21 26 54 1 17 6 6 134 5 1 1 3 5 6 1 9 26 10 1 30 24 3 104. 7 1 9 19 16 18 31 7 101 8 7 15 4 1 3 5 7 42 Totals 35 234 289 151 15 129 101 46 1000 IV. — Mothers and Daughters. 1000 Cases. Mothers, Tints. 1. 2. 3. 4. 5. 6. 7. ■ 8. Totals. 1 5 15 3 2 2 2 2 31 2 7 99 67 29 2 15 23 13 255 3 7 77 111 38 1 26 14 6 280 4 5 22 34 46 2 27 21 7 164 5 2 2 3 1 2 1 11 6 13 27 20 1 35 17 7 120 7 13 21 16 1 19 26 9 105 8 1 5 7 2 1 4 12 2 34 Totals. 25 246 272 153 13 129 117 45 1000 Tables for the Collateral Inheritance of Eye-colour. V*.— Brothers and Brothers. 1500 Cases. First Brother, 1 First. 1 1. 2. 3. 4. 5. 6. 7 8. Totals. 1 8 2 3 4 19 1 2 36 202 23 17 6 4 3 291 3 16 182 209 26 4 2 2 441 1 4 6 36 71 84 7 2 206 5 3 2 1 1 7 f 6 3 56 50 39 34 5 6 193 fc 7 6 37 76 48 1 36 36 2 242 8 4 24 26 18 8 6 15 101 Totals 79 542 460 237 1 96 55 30 1500 T 2 Digitized by Google 140 PROFESSOR K. PEARSON AND DR. A. LEE ON OQ '^ VI*. — Sisters and Sisters. 1500" Cases. First Sister. First. 1. 2. 3. 4. 5. 6. 7. 1 8. Totals. 1 10 2 1 1 14 2 17 147 29 6 10 6 2 217 3 10 136 186 24 9 5 3 373 i 3 75 94 66 1 10 249 5 3 5 2 2 1 13 6 2 57 69 55 5 62 9 2 251 7 4 56 61 52 10 59 49 291 8 2 20 10 7 6 13 26 8 92 Totals 48 496 455 213 23 144 106 15 1500 V^ — Brothers and Brothers. Symmetrical System. Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 16 38 19 10 3 6 6 98 2 38 404 205 53 3 62 41 27 833 3 19 205 418 97 2 64 78 28 901 4 10. 63 97 168 1 46 50 18 443 6 3 2 1 1 1 8 6 3 62 54 46 1 68 41 14 289 7 6 41 78 50 1 41 72 8 297 8 6 27 28 18 14 8 30 131 Totals 98 833 901 443 8 289 297 131 3000 VI^ — Sisters and Sisters. Symmetrical System. Tint. 1 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 20 19 11 4 2 4 2 62 • 2 19 294 165 81 3 67 62 22 713 3 11 165 372 118 5 78 66 13 828 4 4 81 118 132 2 56 62 7 462 5 ! 3 5 2 4 6 11 6 36 6 2 67 78 56 5 104 68 15 396 7 4 62 66 62 11 68 98 26 397 8 2 22 13 7 6 15 26 16 107 Totals 1 62 713 828 462 36 395 397 107 3000 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 141 <S VII, — Brothers and Sisters. 1500 Cases. Brcihers. Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 5 9 18 4 1 37 2 20 163 101 36 28 19 13 380 *5 3 9 98 193 50 37 17 14 418 ■S 4 5 36 49 67 3 28 13 16 217 il ' 5 2 5 1 1 2 2 3 16 6 3 47 41 27 4 42 17 14- 1 195 7 4 34 49 22 3 27 30 19 ! 178 8 1 1 10 7 1 10 8 22 ■ 59 ; Totals 47 399 463 208 11 174 107 ,1 1 1 1500 Table for Assortative Mating in Eye-colour. Vni. — Husbands and Wives, 774 Cases. Husbands, Tint 1. '■ 3. 4. 1 5. 6. 7. 8. Totals. 1 2 13 4 3 1 2 25 2 6 87 42 26 16 13 6 196 1 3 6 56 93 31 1 16 11 6 220 4 4 32 35 18 1 1 15 6 1 1 112 5 6 1 1 1 7 6 1 2 38 27 10 ! 1 12 10 1 101 7 1 5 20 28 7 1 6 12 4 83 8 ! 2 8 8 2 2 4 4 30 Totals : 27 1 254 242 98 4 68 59 22 774 Tables for the Direct Inheritance of Eye-colour. Second Generation. IX. — Paternal Grandfather and Grandson. 765 Cases. Paternal Gramdfather, First. i ! 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 4 10 3 1 3 21 2 7 115 31 20 1 6 13 3 196 3 5 64 109 21 10 22 4 235 4 2 25 40 21 9 13 6 116 6 1 1 2 6 14 32 11 15 5 2 79 7 4 16 16 9 1 11 21 2 80 8 6 5 3 16 6 36 Totals i 22 250 236 83 • 2 66 93 23 765 Digitized by Google 142 PBOFBSSOE K. PEAESON AND DR. A. LEE ON X. — Paternal Grandfather and Granddaughter. 681 Cases. Paternal Grandfather, First. 1. 2. 3. 4. 5. 6. 7. • 8. Totals. 1 3 6 4 5 1 1 ' 20 2 2 94 32 10 2 6 16 4 166 1 3 5 67 71 17 9 20 3 192 4 1 4 36 33 26 1 10 9 3 121 5 3 4 1 1 2 1 1 12 '^ 6 1 16 21 11 15 6 4 ' 74 1 7 3 10 20 11 1 8 15 3 71 8 i 2 5 1 1 1 10 5 25 Totals 18 233 190 82 5 51 79 23 681 Table XI. — Maternal Grandfather and Grandson. 771 Cases. Maternal Grandfather. Tint. 1 i 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 3 11 3 1 1 19 2 8 113 46 22 1 13 5 3 1 211 3 10 87 89 12 11 8 3 220 ^ 1 5 ! 6 4 33 1 25 35 1 25 22 — 15 6 2 117 2 84 2 7 14 7 4 7 1 22 26 6 2 9 10 4 80 8 — 12 12 6 — 4 3 1 38 Totals 1 28 304 237 76 3 66 40 17 771 I' I Table XII, — Maternal Grandfather and Granddaughter. 687 Cases. Maiemal Grandfaiher, Tint. 1. 1 2. 3. 4. 5. 6. 7. 8. Totals. 1 1 1 3 7 2 1 14 2 8 84 35 11 13 6 2 159 3 11 67 76 18 7 15 5 199 4 7 41 40 14 15 11 5 133 6 — 5 2 2 1 — 10 6 4 21 32 1 16 5 2 81 7 4 15 14 8 1 7 19 2 1 70 8 1 5 5 1 1 5 4 1 21 Totals 35 241 211 55 1 61 63 20 ! 687 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 143 Table XIII. — Paternal Grandfather and Grandson. 741 Cases. Paternal Grandmother. 1 Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totak. 1 1 2 7 1 1 2 3 17 2 6 62 69 22 4 15 25 4 207 3 4 31 95 22 1 25 33 9 220 4 3 18 36 20 4 15 16 4 116 5 — — 1 1 1 3 6 1 10 23 6 2 16 10 4 72 7 3 15 10 4 1 14 13 11 71 8 1 10 3 3 — « 5 8 36 Totals 19 148 243 79 13 92 106 41 741 I" I Table XIV. — Paternal Grandmother and Granddaughter. 717 Cases. Paternal Grandmother. Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 3 3 2 6 2 16 2 7 53 56 14 2 13 28 8 181 3 8 35 65 22 6 29 28 7 200 4 4 29 36 20 3 16 23 8 139 5 1 — 3 1 5 1 — 11 6 9 29 2 4 27 8 3 82 7 — 10 15 12 2 10 12 7 68 8 1 2 1 2 1 - 4 4 5 20 Totals 21 141 208 75 18 104 110 40 717 I Table XV. — Maternal Grandmother and Grandson. 756 Cases. Maiemai OrcMdmotker. Tint. 1. 2. 3. 4. 5. j 6. 7. 8. Totals. 1 1 10 1 3 1 3 1 19 2 10 68 53 23 24 13 13 204 3 9 39 67 38 32 23 11 219 4 5 6 3 34 19 1 11 30 1 10 — 19 8 4 117 2 84 ^ 20 z 24 18 1 7 2 9 23 11 1 17 17 — 80 8 — 4 6 5 — '< 6 ' 7 31 Totals 1 25 t 184 181 121 1 122 85 37 756 Digitized by Google 144 PROFESSOK K. PEARSON AND DR. A. LEE ON Table XVI. — Maternal Grandmother and Granddaughter. 739 Cases. Maternal Grandmother. Tint 1 1. 2. 3. 4. 6. 6. 7. 8. Totals. , 1 2 16 18 ;§ 2 7 66 34 13 — 21 15 6 162 §* 3 12 62 55 25 1 27 23 5 210 ^ 4 6 32 36 25 23 15 7 144 i 5 1 3 2 3 3 12 1 6 1 14 21 11 27 17 2 93 7 1 "■" 19 17 7 — 16 17 3 76 8 1 5 4 -2 — 7 3 3 24 Totals 28 212 170 85 1 124 93 26 739 Tables for the Collateral Inheritance of Eye-colour. Second Degree. XVIL — Paternal Uncle and Nephew. 1290 Cases. Paternal Uncle, Tint. 1 1. 2. 3. 4. 5. 6. 7. 8. Total. 1 4 10 11 6 1 4 5 2 43 2 11 136 98 40 26 48 12 371 § 3 8 84 157 26 1 27 54 7 364 ^ 4 29 69 36 1 19 27 12 193 5 2 1 2 1 6 6 1 31 35 7 1 30 19 3 127 7 2 21 27 24 1 13 34 11 133 8 11 7 6 10 8 11 53 Total 26 324 405 145 5 131 196 58 1290 XVIII. — Paternal Uncle and Niece. 1128 Cases. Paternal Uncle. Tint. 1. 1 2. 3. 4. 5. 6. 7. 8. Total. 1 2 10 6 2 1 6 2 29 2 7 85 61 27 29 26 13 248 ^^ 3 1 6 82 126 29 1 26 43 7 319 •^ 4 2 47 73 40 1 29 40 5 237 -e; 5 1 8 1 4 1 5 4 24 6 1 26 35 12 1 8 42 3 128 7 : 1 20 26 19 22 26 7 120 8 1 4 2 3 5 3 6 23 Total 1 26 282 329 136 3 121 191 47 1128 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 145 Table XIX. — Maternal Uncle and Nephew. 1242 Cases. Maternal Unele. Tint. 1. 2. 3. 4. 5. 6. 7 8. Totals. 1 1 8 13 3 3 4 1 33 2 17 137 71 29 19 14 9 296 g 3 10 128 153 26 29 34 3 383 1" 4 2 50 62 28 22 14 1 179 I 5 1 1 2 6 4 33 29 12 35 20 3 136 7 1 33 40 11 26 27 2 140 8 9 17 23 8 3 13 73 Totals 35 399 385 132 142 117 32 1242 i Table XX. — Maternal Uncle and Niece. 1434 Cases. Maternal Uncle. Tint. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 2 15 9 2 2 1 31 2 6 99 76 23 18 13 11 246 3 18 100 108 23 37 36 10 332 4 4 72 64 28 - 16 21 9 214 5 14 2 3 8 5 32 6 5 38 41 10 23 11 4 132 7 1 27 25 7 19 14 3 96 8 15 5 6 9 11 6 51 Totals 36 380 330 102 132 112 42 1134 * Table XXI. — Paternal Aunt and Nephew. 1186 Cases. Paternal Awnt. Tints. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 6 13 5 3 1 4 6 4 42 2 19 113 83 45 36 29 5 330 3 10 81 147 30 — 29 35 8 340 4 8 28 66 38 — 18 22 11 191 5 .i__ __ ._», _ — — ^ 6 3 23 35 12 1 35 10 5 121 7 5 22 28 19 18 16 5 112 8 1 4 9 8 — 6 4 13 47 Totals 52 284 373 155 2 148 121 51 1186 VOL. CXCV, — A. V Digitized by Google 146 PBOFESSOR K. PEAESON AND DR A. LEE ON Table XXII. — Paternal Aunt and Niece. 1149 Cases. Paternal Aunt. Tints. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 2 11 2 3 _ 2 11 2 33 2 16 89 62 37 2 25 40 14 284 3 12 93 119 40 3 41 26 12 346 4 10 36 62 43 5 25 21 11 213 6 5 7 — — 1 3 — 16 6 1 24 33 16 1 29 19 5 128 7 2 20 28 12 2 10 22 4 100 8 — 7 — 4 — 5 9 4 29 Totals 42 28.5 313 155 13 138 151 62 1149 t Table XXIII. — Maternal Aunt and Nephew. 1145 Cases. Maternal Aunt. First. 1. 2. 3. 4. 5. 6. 7. 8. Totals. 1 4 8 7 3 2 3 1 28 2 5 117 81 29 43 29 6 310 3 1 73 132 38 57 43 3 347 4 1 20 54 27 1 21 11 — 135 5 3 2 — 1 — — 6 6 — 24 35 22 — 30 23 3 137 7 — 26 29 20 1 26 25 8 136 8 — 14 6 10 — 12 4 2 47 Totals 11 282 346 151 2 192 138 23 1145 Table XXIV. — Maternal Aunt and Niece. 1058 Cases. Maternal Aunt, First. 1. 2. 3. 4. 5. 6. 7. 8. 1 Totals. 1 2 3 10 15 1 6 87 86 31 — 23 14 12 ; 258 3 3 71 125 32 1 49 41 3 1 326 4 — 39 51 31 1 33 19 6 , 180 5 — 4 6 1 — 8 5 3 ! 27 6 1 25 47 10 — 24 9 2 ' 118 7 — 30 29 11 14 18 10 112 8 — 5 4 5 — 2 4 3 ; 23 Totals 11 264 358 121 2 163 no 39 1058 Digitized by Google MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 147 [Notes added July 3, 1900. Note I. Inheritance of Temper and Artistic Instinct. — In additional to the fraternal correlations given on p. 102, T have dealt with Mr. Galton's statistics for the inheri- tance of good and bad temper given in his 'Natural Inheritance' (p. 235). The following gives the distribution of good and bad temper among 1,294 brethren, as deduced by Mr. Yule. First Brother. I I Co Good Temper. Bad Temper. Totals. Good temper. 330 255 585 Bad temper. 255 454 709 Totals 585 709 1294 The correlation is '3167. A like table is that for artistic instinct in the direct line : — Parentage. Artistic. Non-artistic. Totals. 1 296 173 469 1038 "1 372 666 Totals 668 839 1507 In this case the correlation is '4039. The fraternal correlation is somewhat low. The exact significance of the parental correlation is also somewhat vague, as the parentage is classified as artistic when one or both parents are artistic. But the two tables are very suggestive, they indicate how the new method will enable us to deal quantitatively even with characters like temper and artistic instinct to which it is impossible to apply directly a quantitative scale. With the introduction of a third or medium class, I believe it will be possible to obtain excellent results for heredity from very simple observations, and I have in hand at the present time a large series of observations on collateral heredity based upon such simple classifications. The reader should further consult u 2 Digitized by Google 148 PROFESSOR K. PEARSON AND DR. A. LEE ON Mr. G. U. Yule's remarks on the association of temper and of artistic instinct in his memoir on '* Association/' ' Phil. Trans./ A, vol. 194, p. 290, 1900. Note IL On the Correlation of Fertility and Eye-Colour. — In the course of the present paper I have frequently referred to a probable influence of reproductive selection as the source of the progressive change in eye-colour, i.e., to a possibility that eye-colour is correlated with fertility. I saw from Mr. Galton s tables that in many cases the whole family had not been recorded, probably the eye-colour of the dead or of absentees being unknown. It appeared to me accordingly that it would be impossible to deal directly with the problem of fertility. However, it has since occurred to me that there is nothing likely to give the missing members of families a bias towards one rather than another eye-colour, and that we may simply treat them as a purely random subtraction from the total results. Assuming this, Mr. L. N. FiLON, M.A., has prepared for me tables of father's and mother s eye-colour and of the recorded number of their children. From these* I take first the following results, premising (i.) that I call " light eyed," persons with eye-colours 1, 2 and 3, and " dark eyed," persons with eye-colours 4, 5, 6, 7, 8, i.e., drawing the line between light and dark grey ; (ii.) that I take as small families those with 0, 1, 2, or 3, recorded children and us large those with 4 or more recorded children. Father. Light Eyed. Dark Eyed. Totals. 1 313 Ul 139 280 1 454 403 1 264 577 Totals 857 Mother Light Eyed. Dark Eyed. Totals. ^ M 253 202 455 & 3 225 169 394 Totals 478 371 849 * Correlation tables were prepared of the size of families to 15, and of the eye-colours 1 to 8, but it does not seem needful to print them in ext-enso. Digitized by Google MATHEMATICAL CONTMBUTIONS TO THE THEORY OF EVOLUTION. 149 We have, accordingly, by the method of the present memoir : — Correlation of size of family with darkness of eye-colour = '0595, for fathers. = — -0239, for mothers. The former is- just sensible, the latter hardly sensible relative to the probable error. So far as they can be relied upon, they would denote that fathers have more children the darker eyed they are, and mothers more children the lighter eyed they are. This is in accordance with the result given in the memoir, that the modern generation is darker than its male and lighter than its female ancestry, but it is not the explanation given in the text, although it is probably the true one. If it be the true one, dark fathers and light mothers ought to present the most fertile unions, and it seemed desirable to test this directly. We have already seen that there exists an assortative mating in eye-colour, like tending to mate with like, the CO -efficient of correlation being about '1 ; hence if we were to correlate the eye-colour of mothers and fathers, i.e., husbands and wives weighted with their fertility, we ought to find this result substantially reduced. The following is the table : — Fatliers. I -d Light Eyed. Dark Eyed. Totals. <D >^ P^ 1183 612 1795 ^ bO >A nj a> ,>» W 826 455 1281 ^ u ee Q 1 Totals 2009 1067 3076 We find ?• = '0239, or the correlation has been reduced to a fifth of its previous value, and is now of the order of its probable error. To mark still further this increased fertility of heterogamous unions, I add two further tables, giving the mean number of recorded oflfepring for various classifications of parental eye-colour. Digitized by Google 150 MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. Fathers. Fathers. I Colours 1-3. Colours 4-8. Average of totals. if 3" 3-83 4-57 407 § 00 3-82 3-73 3-79 Average of totals 3-83 417 3-94 ^ Colours 1-2. Colours 3-8. Average of totals 3-86 3" 319 4-52 P 3-98 3-96 3-97 Average of totals 3-68 409 3-94 The first table entirely confirms all the conclusions reached, — dark fathers and light mothers are most fertile absolutely and in union. The second table shows that it is the blue-green and grey rather than pure blue-eyed mothers who are most fertile. This supplementary investigation accordingly seems to support the view of the text of the memoir, namely that reproductive selection is the source of the secular change in eye-colour noted, only the prepotent fertility which is replacing the blue-eyed element is in the first place that of the dark-eyed male, and only in the second place due to mothers having eye-colours dark or light other than true blue. We seem accordingly in eye-colour to have reproductive selection working through heterogamy rather than through homogamy as in the case of stature.* The effect, however, is like, — the progressive elimination of one type of character.] ♦ See * Roy. Soc. Proc.,' vol. 66, p. 30, and vol. 66, p. 316 et seq. Digitized by Google [ 151 ] IV. 071 Simultaneous Partial Differential Equations. By A. C. Dixon, Sc.D. Communicated by J. W. L. Glaisher, Sc.D. Eeceived May 9— Read June 15, 1899. Contents. Pages § 1. Introductory 151 §§ 2 — 9. On " bidifferentials," or the elements of double integrals, and on the conditions to be satisfied in order that a given bidifferential expression may be a complete bidiffer- ential 152—159 §§ 10—13. Theory of equations linear in the Jacobians of two unknown functions; their solution reduced to the formation of complete bidifferentials 159 — 162 ^ 14 — 30. Theory of other simultaneous partial differential equations in two independent and two dependent variables. A method of solution, with examples of its application. One pair of variables is said to be a " bifunction " of other pairs when its bidifferen- tial can be linearly expressed in terms of theirs : this idea is of importance in con- nection with the derivation of all possible solutions when complete primitives are known. Construction of bif unctions in some cases 162—181 ^31 — 42. Differential equations of the second order with one dependent and two independent variables. A method of sohition, with examples 181 — 191 § 1. In this paper, without touching on the question of the existence of integrals of systems of simultaneous partial diflferential equations, I have given a method by which the problem of finding their complete primitives may be attacked. The cases discussed are two : that of a pair of equations of the first order in two dependent and two independent variables, and that of a single equation of the second order, with one dependent and two independent variables. I follow, as far as possible, the analogy of the method of Lagrange and Charpit, and with this object introduce the conception of the " bidifierential " or differential element of the second order, which bears the same relation to a Jacobian taken with respect to two independent variables as a differential does to a differential coefficient. The solutions considered are, in general, complete primitives, that is, such as contain arbitrary constants in such number that the result of their elimination is the system of equations proposed for solution. The existence of such primitives is sufficiently established (see the papers of Frau von Kowalevsky and Professor Konigsberger, quoted hereafler) ; it will therefore be assumed, and the object of the investigation 5.11.1900 Digitized by Google 152 MR. A. C. DIXON ON SIMULTANEOUS will be to find conditions that must be satisfied by the equations of the solution and to put these conditions in a convenient form for solution by inspection. I should add that I am greatly indebted to the referees for their suggestions and for help in removing obscurities. To the list of authorities given by Dr. Forsyth (' Theory of Differential Equations/ Part I., pp. 299, 331), may be added the following : — Julius Konig. Math. Annalen, vol. 23, pp. 520, 521. Leo Konigsberger. Crelle, vol. 109, pp. 261-340.* Math. Annalen, vol. 41, pp. 260-285.t Math. Annalen, vol. 44, pp. 17-40. Ed. v. Weber. Mlinchen Ber., vol. 25, 423-442. J. M'CowAN. Edinb. Math. Soc. Proc, vol. 10, 63-70. Hamburger. Crelle, vol. 110, pp. 158-176. C. BouRLET. Annales de I'ficole Normale (3), vol. 8. RiQUiER. Comptes Rendus, vols. 114, 116, 119. Annales de Tficole Normale (3), vol. 10. Lloyd Tanner. Proc. Lond. Math. Soc, vols. 7-11. J. Brill. Quarterly Journal of Math., vol. 30, pp. 221-242. Several of the above papers are only known to me through abstracts. On Bidifferentials, § 2. The idea of a ** complete differential" plays an important part in the theory of differential equations. In this paper I shall try to show the importance of an extension of the same idea to differential elements of higher orders, such as enter into multiple integrals. An expression Xc/x + Ydy is called a complete differential when X, Y are functions of the independent variables x, y, such that 8Y/ax = ax/ay. If this is the case, then, under certain restrictions, the value of J(Xc/a: + Yo??/) depends only on the limiting values of the variables, and not on the intermediate ones by which these limits are connected, or, as generally expressed, on the path along which the integral is taken. This depends on the theorem that fdY 8X\ \{Xdx + Ydy) = \\(^i-^yxdy * For reasons stated below, I am not in agreement with the results given in the latter part of this paper. t In this paper it should be noticed that the equations (52) on p. 266 are not more general than (46). Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 153 when the single integral is taken round the boundary of the area over which the double integral is to extend. Further, X, Y are in this case the partial derivatives of a single function. § 3. Let us consider the double integral \\(Xdydz + Ydzdx + Zdxdy), where X, Y, Z are functions of the independent variables or, y, z. It is known that this, taken over a closed surface under certain restrictions, is equal to the triple integral J J J(aX/aa; + a Y/ay + dZ/dz) dx dy dz taken over the space enclosed by that surface. Hence, if dX/dx + SY/dy + dZ/dz = identically the double integral taken over a closed surface vanishes, and taken over two open surfaces with the same boundary has the same value ; that is to say, the value of the double integral depends on the values of a;, y, z at the boundary only, and not, under certain restrictions, on the form of the surface enclosed by the boimdary. By analogy we may call the element of the double integral a " complete double diflTerential," or a " complete bidiflferential " \mder these circumstances ; the condition that X dy dz + Y dzdx + Zdx dy may be a complete bidiflferential is thus dX/dx + dY/dy + dZ/dz = 0. § 4. A complete bidiflferential may be expressed as a single term, such as dti dv. For let u^ vhe two independent solutions of the equation SO that n = a, v = 6 are integrals of the system cte/X = dy/Y = d2/Z; then ' X = ^|^, Y = ^|^, Z = ^|^>, 3(y,«)' o(z,x)' d(x,y' 6 being some multiplier, d ax 8Y az_ 3(g,tt,t>) 3lK dy dz d{x, y, z)' Since the last vanishes identically ^ is a function of u, v only ; a function w of tt, v may be foimd, such that Zwjdu = 6, and thus X VOL. CXCV. — A. __ 9(w, v) y _ 9(w, v) „ _ d(w, v) -d(y,z)' '~8(2,r)' ^-a(r.y)- Digitized by VjOOQ IC 154 MR. A. C. DIXON ON SIMULTANEOUS Now in finding the value of the double integral taken over a part of any surface, it will be natural to suppose the co-ordinates of any point of such a surfia.ce to be fiinctions of two parameters, say p, q, and to transform the integral into one taken with respect to these. The integral as transformed is \\{- 9(y.2) + Y 3(g, x), ,j^(x,y) + Z 3(f2)^ a(2>,j)^ d(p, nW"^- and the known values in terms of p, q are to be substituted for x, y, z and their derivatives, ITie subject of integration is dw ^t dio dy dw dz hv dx dv dy^ dvdz 3r 9p 8y 9p 92; 9p ' 9iB 2jp dy dp dz dp dw dx dw dy^ dw dz Sv^dx dv dy dv ^ dx dq dy dq dz dq* dxbqdydqdzdq or The integral is therefore d(Wt v) and if we take a single element we may write Xdydz + Ydzdx + Zdxdy=^ dw cZv, dropping the parameters jp, q^ since the values which x, y, z have in terms of them are immaterial. This equation is meaningless unless the expression in terms of parameters is under- stood. The same is true of ordinary differentials. If when w is a function of sc, y, z we write da^^dx + ^dy + ^dz, we mean that if x, y, z are supposed to be any functions whatever of a single parameter jp, then die du dx du dy ^^du^ dz^ dp '^ dx dp dy dp dz dp' This equation being true quite independently of the expressions assumed for x,y,zm terms of jp, we drop the denominator dp for convenience ; but in modem works on the Differential Calculus it is quite understood that a differential by itself is meaningless apai't fi'om this or some equivalent convention. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 155 § 5. The fiinctions w, v are not uniquely determined. They may be replaced by W, V, where W, V are functions of w, v, one of which, say W, is arbitrary, while V is only restricted by the condition d(w, v) The transformations of ?/;, v which are allowable will thus form a group. For a single integral the operations of the corresponding group consist in the addition of different constants, that is, in varying the constant of integration ; the theory of periodic functions is connected with discontinuous sub-groups of this. It is possible that an investigation of the discontinuous sub-groups of the group of transformations of two variables which leaves their bidifferential unchanged may lead to an extended theory of periodic functions of the two variables. § 6. The finding of the functions w, v may be considered as the indefinite integration of the bidifferential expression. It is simplified by Jacobi's theory of the last multiplier, which is here a constant. we have Xdy — Ydx = ;^ ^^^ "" ^ ^^ > and thus, on the supposition that v is constant, , Xdv-Ydx Ydz-Zdy Zdx-Xdz dw = — ^ = 5 ^ = 5 ov cv ov dz d-x xy _ (fi Z - vY)dx + (i/X -^ \Z)dy + (XY -- /xX)rfg Hence w may be found, if v is known, by integrating this last expression on the supposition that v is constant ; X, /x, i/ may have any values and the constant of integration is to be replaced by an arbitrary function of v. Thus, when one of the functions w^ v is known, the other is found by ordinary integration. The only restriction on the one found first is the equation § 7. Let us now suppose a greater number of independent variables. Let ?^ be a function of aj^, Xg . . . o?^. We have the relation X 2 Digitized by VjOOQ IC 156 MR. A. C. DIXON ON SIMULTANEOUS Here the ditterentials represent simultaneous infinitesimal increments, those of the independent variables being arbitrary. The equation may also be interpreted by supposing «!, X2 . . . x„ to depend in any manner on a single parameter p, when the equation du * 3m dxr ^'P ""r=i3'V dp holds whatever functions of the parameter we suppose x^ . . . aj^ to be. To get the idea of a double differential we must suppose two sets of simultaneous infinitesimal increments ; denote them by d, 8. The bidifferential of a;, y is then dx . 8y -— 8x . dy* This vanishes if a?, y are not functionally independent, just as dx vanishes if re is a constant. The analogy is very clearly shown if we say that dx vanishes when some function i^{x) vanishes, dx dy vanishes when some function ^(Xy y) vanishes. If Uy V are functions of n independent variables x^^ x^ . . . aj», we have dw = S 5- dXr, 8w = 2 5- SaJr ** ov * 3t? di; = S ^ dXry 8t; = S ^- 80:^, and hence du.hv — Zu.do — % S ^ ^ {dXr . hXf — dx^ . 8a:^), or dudv = X ^ ' dxr dx. the summation being taken over all pairs of different suffixes r, s. Hence the expression for du dv is formed by multiplying together 2 5— dXr and 2 ^ cte;. with the conventions dxdy = — dy dx, dx dx = 0. We shall often use the notation d{x, y) for dx dy. § 8. For the purpose of double integration of such an expression as 2 X„ d{xr, a?,), in which the coefficients X are ftinctions of acj . . . a:„, it is natural to suppose a:^ , . . a;» expressed throughout the range of the integration in terms of two parameters, say jp, q. The integral thus becomes * The dot is used here and throughout the paragraph to distinguish multiplication in the ordinary algebraic sense from multiplication according to the Grassmann conventions stated at the end of the paragraph. Digitized by VnOOQ iC PAETIAL DIFFERENTIAL EQUATIONS. 157 w If X„ = ^ * \ for all pairs of suffixes, the subject of integration in the last integral is d{tf^ v)/d{pf q)^ so that the. integral becomes Hdudv. Its value will therefore only depend on the values of w, v, that is of jc,, a;^ . . . x^, at the boundary of the range of integration, and not on the form of the relations giving 0:^, x^ . . . in terms of ^, q^ which define the particular siu'face over which the integral is taken. In this case we may write 2 Kr, d{Xry X,) = d(u, v) and call it a complete bidifferential. It is easily seen that the coefficients X satisfy the relations X,,X(,. + XhX,, + X^.X« = Oi ...... . (1) .' -t+t+t-" -f^'' for aU combinations of suffixes, where it is understood that the term Xr,c?(a:^, x,) may be also written X^d{x,^ Xr), so that Xf , = — X,.,. The conditions (2) are those which must be satisfied in order that the value of the double integral may depend only on the boundary. The difierence of two values of the double integral, for which the same boundary is assumed, will be its value over a closed surface passing through the boundary curve, and this may be transformed into the triple integral fffi,(t + l: + th^-^^- taken through the voluriie of any solid bounded by this closed surface. Hence this integral must vanish for any solid. By taking an infinitesimal solid, for every point of which all but a;,-, Xr, Xg are constant, we find the condition (2). The conditions (2) would be satisfied by an expression which was the sum of two or more complete bidifferentials, but (1) in general would not. § 9. We next try to find whether these c6nditions are sufficient as well as necessary. Now all the coefficients X cannot vanish. Suppose that X^g does not, then we have firom (1) and in virtue of these all the conditions (1) are satisfied. Digitized by VnOOQ iC 158 MR. A. C. DIXON ON SIMULTANEOUS Taking the values thus given for Xr„ X,>, X«- we have ~ x,jl aPi . ■*" eb-,/ ■•■ XijV &•.• ■•■ -ar,/ "^ x„V a^, "^ a^. / Y /^ J. ^^ J. ^'Y^ J- ?^\ 4- ^?t/^' -U ^^ x„ axjg Xfr axjg Xjj axjg X]2 a^'i Xjg aa7j Xj3 atv = 1^ (IW) + ^ (r2i) + 1^' (2st) + I" (sit) + |y («2r) + ^ (1^) -^13 -*^12 ^12 ^12 -^12 -^12 M2*^2 + 1^ (21«) + ^ (21r) + 1^ (21t). Ajj Aj, -a.12 Thus the conditions (2) are not independent, but all follow from those in which at least one of the suffixes 1, 2 enters. If they are satisfied then the equations H n S Xi^ dXr == 0, 2 Xgy dXr = rss2 r=l can be satisfied by two integrals of the form w = a, v = h; that is, these last equations will give Xj, ic^ as functions of the rest, such that &i Xg^ ^ Xrt For the conditions necessary and sufficient* for this are the vanishing of such expressions as J^_ ?!l 4. ?«? . ^_ ?!1J I ^1 ^ ^1 _ 9 Xrt ^ Xe^ j^ Xrt ^ Xyi 9 Xrt d^, Xjg Xjj &i Xi2 Xjg OTg Xjj 9.ZV Xj2 Xj3 Oit'i Xij Xjj ocj X^ in which 1, 2 may be interchanged and i% s are any two of the other suffixes. This expression may be written i;, <^l*) " §; (^2s) + || (12r) + ^ ^ (Xi2 X„ + X,, X^ + X^ X,.), SO that it vanishes and the conditions of integrability of the equations XXj^Xr = 0, r l,X2rdXr = are satisfied. If w = a, v = 6 are the integrals, then, since tK^dxr r does not contain the diflferential of x^, we must have * For proof of this statement see Fobsyth, * Theory,* part I., pp. 43-61. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 159 d(u, v) and Xir = 3(«i, av)' In like manner Xgr = 6 ^-7 — :, the multiplier being the same. Hence ■- ^ = *^! Since this vanishes for all combinations of suffixes, ^ is a fimction of u, v, and if another function of them, ti;, is so chosen that we shall have t X^, d{Xr^ x,) = d{Uy v) = d{w, v). r,« Linear Differential Equations. § 10. If t^ = a is an integral of the linear partial differential equation where X^, X^ . . . X^^.! are ftmctions of x^y . . . x^^^i^ then n satisfies the condition S X,^ = 0, r=l OXr and the complete differential du is a linear combination''^ of the determinants dxiy dx^i dx^ • • . dx^y dXf^+i ^l9 -^> -^ • • • -^J -^+1 the coefficients in the combination being usually functions of x^, . . . Xm^i. If u = a is a conunon solution of the above equation and of ^1 ai^i + • • • +^ Stew --^-+^' then, in like manner, duiaa, linear combination of the determinants * This is generally expressed by saying that *' it = a is an integral of the equations Xi X2 Xn+i For the sake of the analogy with the work of § 11, I prefer the phrase in the text, which expresses no more and no less than the one generally used. Digitized by VnOOQ iC 160 ME. A. C. DIXON ON SIMULTANEOUS dxu ^^2> • • • r^n+u -^l> ^> • • • J -X-n+i, X' Y ' Y' but in general, of course, it will not be possible to combine them so as to form a perfect differential. § 11. An analogous process of integration may be given for two simultaneous equations 2{Aj,(My - Pjqi)} + SB.?>.- + tC-qi + E = 01 t{A:^{piqj-pjqi)} + tB\'Pi + tC',qi + -E'^0} * * * * (3)' in which the coefficients A, B, C, E, A', B', C, E' axe functions of n independent variables, ajj, x^ . . . x^y and two dependent y, 2, and To fix the ideas, take n = 3 and let x^ x^ stand for y, z respectively, A;4 for C„ Aj5 for — Bj, A45 for E, and make similar changes in the accented letters. Then, if tc =z a^ V = h are two equations constituting a solution,* a, h being arbitrary con- stants, we must have IJ = 1,2..8 'd(a.bXj) (4); and the values thus given for p^^ g^, j?2> Q'sj i^8> Q's ^aust satisfy the equations (3) identi- cally, since a, & are supposed arbitrary. The equations to be solved are thus reduced to others which are linear and homogeneous in the Jacobians, and which do not contain the dependent variables. The equations (4) give two of the Jacobians of a, v linearly in terms of the others ; if we substitute tor these two in the identity '*(''.'')=a^)'^^-A we find that d{u, v) is a linear combination of the determinants of the matrix of tent columns. * This solution will not be a complete primitive unless a certain number of other arbitrary constants are involved as well as a, 6, a supposition which is neither made nor excluded. It may be well to point out that the solution here assumed consists of two equations, and not of one equation involving an arbitrary function ; in fact, any solution whatever necessarily consists of two equations, and one point of the present method is that these are to be sought together, not successively. t For n independent variables the number of columns in the matrix will be |(n + 1) (n + 2), the number of rows being still thiee. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 161 d{x^, x^), d{x^, a,), . . . d{xi, Xj) . . , d{x^, x^) A' A' A' A' There are thus eight bidiflferential expressions, and the problem is to be solved by finding such multiples of these as, when added together, will form a complete bidifferential. § 12. As in the case of Lagrange's linear equation, this will generally, in practice, be done by inspection, and the method will be useful for finding solutions in finite terms — when such exist. But in any case,* whether the inspection is successful or not, there can be no doubt of the existence of suitable multipliers, in infinite number. For it is certain that the equations (3) have — possibly among other solutions — an infinity of solutions, each involving two arbitrary constants at least, and any one of these may be written t^ = a, v = 6, where a, h are the two constants ; m, v are functions of the variables, but may, of course, be implicit functions of great com- plexity. The functions w, v must satisfy the conditions (4), and it immediately follows that d{u^ v) must be a linear combination of the determinants of the matrix formed from (4) as above ; so that a corresponding system of multipliers must exist. If the solution is not in finite terms it is not likely to be found by inspection, and it is quite probable that the best way to find it would be by solving the original equations (3) in series. By whatever means the solution is found, the corresponding system of multipliers is thereby determined. If nine solutions of the form w = a, v = 6 have been found, the nine bidifierentials rf(wi, Vj), ^(ttj* Vg) . . . d{ug^ Vg) must satisfy identically a linear rela- tion, since they are all linear combinations of eight expressions only. We shall say that one of the nine pairs of functions is a " bifunction " of the other eight pairs. The following is, then, the definition of a bifunction. When the bidiflferentials of any nimiber of pairs of quantities are connected by an identical linear relation, with constant or variable coefficients, any one of these pairs is said to be a bifunction of the rest. The word bifunction is simply used as an abbreviation — at least for the present. I am not without hope that at a future time it may be found to have some connotation. * If one of the dependent variables with its derivatives is altogether absent from the equations (3), or if it can be made to disappear by a change of the other dependent variable, the equations (3) will in general have no solution. This case will then be excluded ; it is the only case in which the method of solution in series (as given, for instance, by Frau von Kowalevsky, * Crelle/ vol. 80) camiot be used to prove that solutions actually exist. Another case that may fairly be excluded is that in which all the derivatives of one of the dependent variables do not occur or may be made to disappear by a change of the other. Such a system is equiva- lent to a single partial differential equation with one dependent variable, since the one whose derivatives are absent may be eliminated. VOL. CXCV. — A. Y Digitized by Google 162 MR. A. C. DIXON ON SIMULTANEOUS It is, of course, evident that if % v are functions of variables a^i, x^ . . . then the pair u, V 18 a. bifunction of all the pairs that can be formed from ajj, x^ , . . Other examples will be found later on in the paper. § 13. Sometimes solutions exist for systems of partial differential equations in which the number of dependent variables is less than the number of equations. If, for instance, with the system just considered we take a third equation of the same form, the coefficients being distinguished by two dashes, there may be solutions common to the three equations. If u — a^ v = h give such a solution, then it follows in like manner that d{u, v) is a linear combination of the determinants of the following matrix : — d{x^,x,) . . . d{xi, xj) . Ai2, . . . A,y, . . A' -^ 12> • • • A'.. A'' A.% : . Similarly for a greater number of equations. Application to other Differential Equations. § 14. There are two classes of equations whose solution depends on that of a pair of such linear homogeneous equations as we have just been considering ; they are, firstly, systems of two equations in two dependent and two independent variables, and, secondly, equations of the second order with one dependent variable and two independent. We shall consider them in order. Firstly, let y, zhe the dependent variables and .Tj, a;., the independent ; sometimes we shall write x^ for y and x^ for z. Let Pi^ p^ be the partial derivatives of y and g^i, q^ those of a;, and let the equations be /IC^I. ^2. Vy ^y Pu Pzy ?1> Qz) = 0, Ai^v ^2. y> 2;, ^i, ^2, g'l, q^) = 0. A complete primitive will consist of two equations connecting x^y x^, y, z and involving four arbitrary constants. By differentiation these equations yield four more involving p^, p^y g'l, g'2- ^^ t^® ^^^o equations are supposed to be a complete primitive it must be possible to find expressions for the four arbitrary constants in terms of aj^, x^, y, 2, Pi, 5^1, P2> qz I t^® elimination of the four constants must give /i = 0,/2=0. Let «!, ag, ag, a^ be the constants, and ^i, t^o, w,, u^ the expressions for them in terms of x^, Xc^y y, 2, p^, q^y p^^, q.^. Suppose /g, f^yf^.f^ to stand for u^y u^y Vg, %i^ respectively. Then by differentiation we have for any value of the suffix i from I to 6, Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 163 the letter d being used to denote differentiation with respect to x^ or Xo on the sup- position that the other is constant, while 3 indicates strictly partial differentiation. Since dpjdxx = dpjdx^, dqjdx^ = dqjdx^^ we find by eliminating the deriva- tives of jpi, gi, ^2, g^2, that J(^i>i>i. 9^1. 5^2) +i>iJ(y.i>i> <iu 92) + qi^^y Ply qu ^2) + J(^2>P2> 9i> 92) + Pr^iVy V^y 9\y 9%) + 92^{^yV2y 9\y 9%) = 0, and %!, ji, ^1, p^) + ^1%, ^1, p^, p^) + q^J{z, q^, p^, p^) + J{x^, q^, p^, p^) + P2^yy 92y Ply Pi) + 92^{^y ?2>Pl>i^2) = where J( ) denotes the Jacobian of any four of the functions /j, f^yfzyf^^fhyA ^*^^^ respect to the variables specified in the bracket. Of these equations there are thirty, but since they are given by the elimination of six quantities from twelve equations only six of the thirty can be independent. § 15. One pair of these auxiliary equations will contain Jacobians of /i, /g, ^, f^y and will in fact express the conditions that the equations dy = p^dxi + p^dxc^ dz = q^dxi + q^dxc^ shall be integrable without restriction when p^, p,^, q^, q^ have the values given by the equations /i = =fc^, /g = a^,f^ = a^. Thus, if a pair of functions ^, J^ can be found satisfying these two auxiliary equations, the solution can be completed by solving a pair of simultaneous ordinary equations. (See Mayer's method, Forsyth, * Theory of Differential Equations,' pp. 59-62.) The two auxiliary equations that^^ f^ must satisfy are linear and homogeneous in their Jacobians, the coefficients of the Jacobians not involving the functions f^, f^ ; the number of independent variables is apparently eight, but it may be taken as six, since two of the eight variables x^, Xg, y, z, P\y p^y 9\y 9^ ^^^ given as functions of the other six by the relations /j = 0,^ = 0, and may be supposed eliminated from^j,/^, if that is desirable. The colimins of the matrix formed as at § 11 are the rows of the following array : — c?(xi, Xg), 0, 0, d{x^y y\ 0, 0, d{x^y z\ 0, 0, d{xzy y)y 0, 0, (5) d{xz, z), 0, 0, Y 2 Digitized by VnOOQ iC 164 MR. A. C. DIXON ON SIMULTANEOUS %,«), 0, 0, ^{^u Pi). (91. 93} » {Pi,qi}> d{x^,p{), 0, {Pi,q'i], %. Pi)>Pi{qv %} pAps, 9i} + PiiPi, qi), (10) d{z, Pi),gi{?i, q,], qAPi, qi] + q^iPi, q^], (^i^!lyPi)>{q^yqi}> {q2,pi], ^iy'Pi\Pi{quqi]> PiiquPi) +Pi{qiyPi]* d{^>P2)> qdqu qi}y qi{q\>Pi) + qi{qi>P))> d{yi,qi),pi{qci,Pi] •\-P2{qi,Pi}, PifPi.i'al. c^(2.?i). ^ifo'z.i'i} + qi{qi>Pi}, qi{px.Pi}> d{xi,qi),{px,qi}y 0, . (20) d{Xi, q^), (pa, q^}, f i>i, P2} , %. 92), I'll;?!, qi) + P2(P2. g'l}* i'2{pi.i>2}. «^{«. 72), g'lfPl. fZi) + 92(l>2. 9l}. 92{2'|.i>2}. ^0^1. M 0. {a^i, 9i} +i>i{y. 9i} + 9i{«.9i} + {a'2. 92} +Pi{y> 92} + ?s{2. 92}. c?(Pi. 9i). {^\,qi\ +Pi{y, 92} + Q'll^. q%}> -{^i>Pi} -pAy^Pi) - g'j2.i'2}> (25) ci(pi,q.2), - {Xi.^ri} -Pi{y,9i}-9i{2.9i}. -i^i^Pa) -PziV^Pi) - 92KP2}. %2. 9i). W 92} + P2(y.9a} + qd^^qi), {aJi.Pi} +2>i{y.Pi} + gJ^^Pil. ^(P2.</2).-{a;2.!?i}-P2{y.'?i}-?2{2.g'i}. {3^2,^1} +i?j{y,i?,] +g2{2,i>i}, %i. «?2).{a^i. Pi) +Pi{y>Pi] + 9i{2,pi} + {a'2.i>2} +Pi{yy P25 + 92(2. qi}> 0, (5) Here {p^, q^}^ for instance, is written for 3(/i,^)/3(Pi,g'i), and eveiy fifth row is numbered. § 16. In order, then, to solve the equations /^ = 0,^ = we have to form such a linear combination of the determinants of this array as will be a complete bidiffer- ential, say <l{f^, f^, f^, f^ being such functions that the equations /^ = :=^fc^^f^ = aj, f^ = a^ can be solved for jOj, q^, p^, q^. The array contains twenty-eight rows, but thirteen of these are combinations of the other fifteen. For instance, multiply the first row by dfjdx^, the second by dfjdj/y the third by S/J/S^, the seventh by 3/i/3pi, the eleventh by dfjdp^y the fifteenth by Sfjdqiy the nineteenth by 9/i/9g2 a^d add ; the resulting row is ^(^i/i), 0, 0, which vanishes. Other vanishing rows may be formed similarly by combining the rows of the array so as to have in the first column one of the following — (^i^iJll d{x.^J\)> d{yj^\ d{z,f^\ d{p^Jl)y d{p^yfl)y d{q^Jl)y d{q^yf\)y %i>/2)> d{^^2yA\ %>/2)> d{zj^\ dip^yAl d{p^J^\ d{q^J^\ d{q^J^). Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 165 The coefficients in these combinations are partial derivatives of f^ or ^, thus, for instance, d^vM = ^ rf(Pi. ^.) + ^ d{p,, X,) + I rf{^i, y) + I rf(;>„ z) and so in other cases. The number of these combinations is sixteen, but it is to be lowered by three, since d(f^^f^ and d{fci,f^ are identically zero and (^{fi,/^) can be formed by com- bining the sixteen in two ways, so that three linear combinations of the sixteen bidiflferentials vanish identically. Hence the array contains virtually only fifteen rows (28 — 13) and as there are three columns, we have thirteen bidifferential expressions to combine. Any pair of the four Amctions ajj, ^o, y, z will satisfy the two auxiliary equations, as is clear either from the equations themselves or from an examination of the matrix ; of course these solutions of the auxiliary equations will not give a complete primitive. § 17. If a complete primitive has been found it leads, as has been explained, to four equations Ui = «!, Wjj = a^y ^^3 = ag, u^ = a^, and any pair of these must satisfy the auxiliary equations. Thus twelve pairs of functions satisfying these are known, namely a:,anda:^ {ij= 1, 2, 3, 4) u, and Uj {i,j = 1, 2, 3, 4). 'These, however, are not all independent, but one pair is a bifunction of the other eleven. For if <^(a?i, x^, x^, x^, a^, a^, ag, a^) = <^(a?i, X.2, x^, x^, tti, flo, ag, aj = 01 ^{x^, 0^2, ajg, x^y a^, ag, ag, a^) = oJ (6) are the equations of the complete primitive, they must reduce to identities when t/i, U.2, u^y u^ are substituted for a^, ao, ag, a^ respectively. Hence 4>{^ly ^Z. ^3» ^J» '^l' ^2> '^3> ^-^i) — ^ \ /yx ^{x^, a-a, ajg, x^y Uj, W2, u^y W4) = j Identically, and ^ 34> ^ = . ^ -^* du,, t^dxi = — 2 >.* diHy i oxi i aui and the bidifferentials of the twelve pairs of functions are connected by a linear relation. Digitized by VnOOQ iC 166 MR A. 0. DIXON ON SIMULTANEOUS § 18. The method of Charpit for a single partial differential equation of the first order shows how all solutions may be deduced from one complete primitive, and it is a question of interest and importance whether there is any analogous method for simultaneous equations. Now it follows at once from the conditions for a complete bidifferential that a bifunction of the pairs that can be formed from m functions, say u^, Wg . . . . Wm, will be a pair of functions of u^ . . . . Um- In the present case a bifunction of the six pairs that can be formed with u^y t/^, t/g, u^ will be a pair of functions of these four, and the complete primitive to which it will lead will be the same as that given by t^i, w^. For when a solution of the auxiliary equations is known it leads directly to one and only one complete primitive by the integration of the equations* ^y = Pi ^^1 + Pz ^^2 dz = q^dx^ + 9^2 ^^2; also the complete primitive to which the equations F^ {u^y t/g, u^y u^ = const., F2 (ti], t/2, ^^3, u^ = const., will lead can be no other than is given by u^ = ai, 1^2 = ^2» ^'3 = ^s» "^4 = ^4- It must not, however, be forgotten that the system F^ = const., F2 = const., y^ = 0, /2 = may have a singular solution. If F^, F2 involve two other arbitrary constants this singular solution will involve four, and therefore in general be a com- plete primitive of the equations yj = 0, ^ = 0. Moreover, all new complete primitives are included among those thus given. For every solution implies six equations connecting x^^ 0^2, y, z^ Piy qi, p^y q% (two of these six are of course f^ =: 0, f^ = 0), and, therefore, by elimination of ^1, ^2y y^ ^> Pi> 9^1 > i^2> ?2> ^^^ equations or more connecting Mj, Uc^^ %, w^, which are known in terms of these eight quantities. If Wj, u^^ u^y w^^are connected by four equa- tions they are constants, and the solution is therefore included in the old complete primitive. Let us, then, suppose that w^, Wg, u^y ^4 are connected by two or by three equations, F>i, W2, 7^3, u^) = (a = 1, 2 or 1, 2, 3). Now if pi, pc^y g^i, g^2> are all expressed in terms of jo, q, two of their number, and ^i> ^2j 3/> ^y ^y ineans of the equations /^ = 0,y^ = 0, the expressions ^y "" Vidxi — pc^dxc^y dz — q^dx^ — qc^dxc^ must both be expressible in the form K^dui + A^du:^ + Agdwg + A^du^y * Otherwise thus — ^if in the auxiliary equations we suppose /s to have the known value Ui, they become a pair of linear equations for /4, which must be satisfied by u^, Us, U4 ; now two linear equations in six independent variables can only have four functionally independent solutions, and one of these is known, namely, Wi. (In exceptional cases the two linear equations for Wg, ih, u^ may be equivalent ; for instance, suppose fi = pi + qiy Ui = p2 + <tiyh having any form.) Hence, except in special cases, the particular complete primitive is defined when one of the "functions %\y t*2, t^, w^, or more generally a combination of them, F (Wj, w^? ws, W4) is known. In the case supposed in the text two such combinations are known. Digitized by Google PARTIAL DIFFERENTIAL EQUATIONS. 167 and since dp^ dq are absent we must have in each case 2 A,v: =0, t A,z- = 0. r=I op r=l Oq Thus.the equations dy = p^dx^ + pcidxo^ dz = q^dx^ + g^^^aj^ become 1 dii^y du^i du^^ du^f dui Bi/g du^ du^ ^' a^' 3^' 3^ = 0. 3^1 3m2 9^8 du^ a^' 3^' a^' 3^ These two equations, connecting dui^ dii^^ du^^ du^y taken with the system * 3F 2 ^ dur = (a = 1, 2 or 1, 2, 3), show that if Wj, w^, 7^3, U4 satisfy by themselves no other relations than F« = (a = 1, 2 or 1, 2, 3) we must have, as a consequence of the equations of the solution, r=l dlCr 3p ' r=l 3Wr S? If, then, there are two equations Fi = 0, F2 = 0, the four equations S|^|'=0.S 1^1^=0 (.= 1.2) must reduce to two only. This will be the ordinary case, and we see that if the forms of Fi, Fg, have been found by any means, the solution is completed without integration ; the process corresponds to Charpit's method of deducing all complete primitives from one, but it diflfers in that the functions F^, F^, are not arbitrary ; they must, in fact, be so chosen that the four equations last written shall reduce to two, and the conditions for this are clearly very complicated in general, though in particular cases available forms for Fj, F^ may be seen on inspection. In the more uncommon case, when there are three equations F, = 0,F„=0,F3=0, the six equations ii 3;^ V "" ' ii 3z.. "3^ - (a - 1, 2, 3), must reduce to one only. These two cases are further discussed, from a somewhat diflferent point of view, in §§ 21—23. It should not be forgotten that the form in which tlie new complete primitive has Digitized by VnOOQ iC 168 MR. A. C. DIXON ON SIMULTANEOUS just appeared is not that in which complete primitives were discussed in § 14, since the equations are not here supposed to be solved for the arbitrary constants. § 19. In addition to the six pairs {u,, Uj) of functions satisfying the auxiliary equations, we have also the six pairs (oc,-, Xj) ; of these twelve, eleve i are indepen- dent, the other being a bifunction of them. If we can find : bifunction of the eleven pairs which is not a bifunction of either set of six it Avill give a new complete primitive ; whether every, or indeed any, other primitive is thus given is a matter for further inquiry. Suppose Vi = hi {i = 1, 2, 3, 4) to be a new complete primitive, then it gives six more pairs of functions satisfying the auxiliary equations, and thus we have in all eighteen pairs. The bidifferentials of these must be connected by (18 — 13) five linear relations, one of which has been written (8) ; by means of the other four, an expression of either of the following forms — Ad(v2, t's) + Bd{v^, Vi) + Crf(t'i, r.), can be found which will be equal to a linear combination of the twelve bidifferentials d {xiy Xj) and d {uiy uJ). It is natural to ask whether, conversely, any linear combina- tion of these twelve which can be written in one of the above forms will lead to a complete primitive ? In the first case this is not so, for if we take any fiinction whatever, tj, of six independent variables, fj, ^^ . . . fg, we may choose the coefficients <*!>••• «6> so that shall be a linear combination of eleven* given bidifferentials ; the expression t, oLi d^i may then be reduced to three terms, P^ dt,^ + jSg dt^ + ^Sg djg, so that for an arbitrary function {t}) a combination of the eleven given bidifferentials can be found of the form fi^ d{r)y £]) + ^82 d{yi, t,^ + ^Sg d[yiXz)> which is the same as Ao?(i;i, v^ + Brf(t;|, Vg) + Cd(vi, V4). This argument does not apply to the second form Ad(t?2, ^^s) + M^3. '^i) + C!c/(vi, v^\ and further investigation may show that any combination of the eleven that can be reduced to this formf will lead to a primitive. * Not of any lower number in general, since the most general bidiiferential expression in this number of variables contains fifteen terms, while the expression just written vanishes identically if so that there are virtually only five coefficients, of which one must be left arbitrary. t The conditions necessary that a bidifferential expression may be reducible to this foim include algebraic ones which are the same as for a complete bidifferential, since Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 169 § 20. Before we can claim in any sense to have found the general solution of the auxiliary equations, we must be in possession of thirteen pairs of functions satisfying them ; we have only eleven when we know one complete primitive, and hence one more complete primitive, or even possibly two, must be found. An example (below, § 29) will show that one more is not always enough. It is perhaps worth while to remark that any complete primitive defines the whole system of solutions, since it defines the differential equations. § 21. The question of finding new solutions when a complete primitive is known may be attacked by the method of varying the parameters. Take the equations (6) or (7) of § 17. The problem is then to find such variable values for Wj, u^y Wg, u^ as will satisfy the equations S P^dui = 0, 2 ^^dui = (9). Since all variables are supposed functions of aj^, cc^, we may make one of two suppositions with respect to w^, t^g, u^, u^ ; either they are connected by three relations and are all functions of the same variable, say ty which is ot course a function of ar^, Xg, or they are only connected by two relations, so that two of them may be taken as functions of the other two. Suppose first that they are all functions of the one variable t Then, generally, the four equations (7), (9) will define a;^, Xg, ocg, x^ also as functions of ^, and hence this supposition is not admissible unless it is possible to choose the functions of t in such a way* that the four equations (7), (9) will be only equivalent to three. The If these conditions are satisfied by an expression it can be put in the form 2 Ai//(a'i, Xj), t,; = 1.2. . 6 and then it must further be possible to express 6 8 2 ki dxi and 2 /i^ dxi i=l i=l as linear combinations of three differentials, dvi^ dv^y dv^ The discussion of the conditions therefore belongs to the theory of the reduction of two such expressions, that is, of the extended Pfaff problem. * It seems obvious that this will not generally be possible ; but it may be well to give an example. Suppose the complete primitive to be y = axi^ + bx2 + c, ^ z = cxi + ex2^ + bxiX2^j^ so that the differential equations are y = ipi^i + p^ + qi - pix^, z = qiXi + ij2a"2 ~ P^iX2\ then the variations of the pammeters a, &, c, e must satisfy the equations, Xi^da + xodb + dc = Xidc + x-^de + XiX-^dh = 0; VOL. CXCV. — ^A. Z Digitized by Google 170 ME. A. 0. DIXON ON SIMULTANEOUS number of conditions, which will be of the nature of ordinary diflferential equations, thus imposed on the four parameters must not be greater than three ; for if they are subjected to four conditions they are made invariable ; it may be, however, less than three. For instance, a complete primitive of the equations jpi=:p2» 9i = ?2 is given by y = a{x^ + Xg) + 6, 2; = c{x^ + 3^2) + e ; the equations given by varying the parameters are {x^ + Xc^)da + c?6 = 0, (xj + X2)dc + de = Oy which give the single differential equation connecting the parameters da de = db dc. We may then assume arbitrary forms for two parameters in terms of a third, and find the fourth by integration. Say, for instance, b = <f>{a)y c = V^(a), then e = J<^'(tt)i//(a)da, a^i + a:^ = — it>\a) ; thus we arrive at the known general solution y = X(^i + ^2)> ^ = ^(^1 + ^2)- whence, by elimination of xi, {x^Hh + dcy{x^b + dc) + x^iHe^da = 0. This equation must fail to define 0^2, so that 5, c, and a ov e must be constant ; thence it follows that all four parameters must be constant. I lay stress on this, because it is not in agreement with the results of Professor Konigsberger (' Grelle,' vol. 109, p. 318), and appears in fact to show that his method there given is faulty. Professor Konigs- berger assumes (p. 313; I take m = 2) that the most general integral of the equations f\{^\y a^2, y, z, pi, JP2, qu ^2) = Mxu X2, y, z. Ply p2y qi, q^) = has the form y = wi(ari, iC2, </>i[^i(a^i, iCa)], <h[h{^u ^2)]) z = ft>2(ici, X2, </>i[^i(a^i, 0:2)], </>2[^2(a:i, a-^i)]), where </>i, <^ denote arbitrary and ^1, ^2 definite functions. But suppose these equations solved for <^, <^2 iu the form ^\\^\{^\y «2)] = Xi(^i» ^h y, ^) H^i(xiy »2)] = X2(«i, ^y y, ^) and the arbitrary functions eliminated by differentiation. The differential equations thus formed are of the first degree in |?i, ^2, S'l, ?2» *^^ *^® ^^^ V ^"7 n^eans of the general form assumed. The differential equations in the examples given by Professor Konigsberger are, in fact, linear (see pp. 319, 328). The method appears to be founded on an interpretation of the last clause of § 2 (p. 290), which is not justified. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. I7I In the case of two equations of Clairaut's form y = Pi^i + Ih^^ + <^(Pi, i>2. 9^1. 92). z = q^x^ + q^x^ + y^fip^, p^, q^, q^\ which will be more fully considered later, the number of differential relations among the parameters is twOy so that one parameter may be taken as an arbitrary function of a second, and the other two found in terms of the second by solving two ordinary differential equations. If the primitive* is y = aa + 6)8 + cy + ^8 ^ = Aa + B^ + Cy + E8, where a, 6, c, e are the parameters. A, B, C, E known functions of a, 6, c, c, and ^y Py y> 8 known functions of x^, Xc^, then the variations of the parameters must satisfy the relations cda + fidb + ydc + Sde = 0, cudA + fidB + ydC + SrfE = 0, and thus, in general, if a, &, c, e are all functions of one variable they are connected by three relations dA/da = dB/db = dC/dc = cflE/de. The integral equivalent of these equations consists of three relations connecting a, 6, c, e with three arbitrary constants, and by eliminating a, 6, c, e we find a new solution of the original differential equations which is not a complete primitive, since it only contains three arbitrary constants. These examples show that the number of conditions to be fulfilled by the para- meters when all four are taken to be functions of one of them, may be one, two, or three ; this number is to be made up to three by assuming arbitrary relations (two, one, or none, as the case may be). § 22. Usually the parameters will not be functions of one variable only, and we may suppose two of them, Wg, u^, to be functions of the other two, w^, u^. The partial differential coefficients du^ du^ du^ du^ dill * du^ ' du^ ' dtt^ are then given by the equations (9), each of which is equivalent to two. The first, for instance, gives 3<^ 90 duj^ dcf) du^ dui dic^du^ dti^^dtii ' 3^ 9^ du^ d<f> du^ 9^2 du^du^ dic^du^ The derivatives are thus given in terms of le^, w^, Wg, u^, iCj, x^y ajg, x^, and the last * It is imnecessary to give the differential equations. z 2 Digitized by VnOOQ iC 172 MR A. C. DIXON ON SIMULTANEOUS four may be eliminated by means of the relations (7) ; so that in the end we shall have two relations connecting w^, u^^ Uo, u^, and the derivatives ; the problem is of the same form as the original one, to solve two simultaneous partial differential equations in two dependent and two independent variables. Interchange of Variables and Parameters. § 23. A curious thing may be noticed at this point. If in the equations <^ = 0, ^ = 0, we treat x^, x^^ Xz^ x^ as arbitrary constants and eliminate them by differentia- tion, we are led to the same differential equations connecting w^, Wg, W3, u^ as were just now given by the variation of parameters. Thus two equations in two sets of four quantities will give two pairs of simultaneous partial differential equations by taking each set of the quantities in turn as variables and the other as arbitrary con- stants. The auxiliary equations, if expressed in terms of the eight quantities, will be the same in both cases ; this gives a meaning to the six solutions of the form (x,-, Xj) which we found the auxiliary equations to have, for any one of the six wiU lead to the primitive <^ = 0, ^ = of the second pair of differential equations, just as a solution {ui, uj) leads to this primitive for the first pair ; any new solution of the auxiliary equations will in general lead to a new complete primitive for either pair, but an exception to this rule will arise when, for instance, the x differential equations have a complete primitive which gives three relations among u^, u^^ W3, m^. The array (5), transformed so that the variables are aj^, x^, ajg, x^, u^y Uc^, M3, u^, connected by the equations ^ = 0, t/r = 0, will have six rows of the form d{xi, xj), 0, 0, six of the form ^(w/, w^), 0, 0, and in the other sixteen there will be d{xi, uj) in the first column, in the second the minor of ^ ^ in the determinant : ^<f> d^<f> S?<f> 3^^ 30 dyjr ctejBMi ' d^idu^ * ari8?tj ' dxjdu^ arv a.-i d'-4> d';f> d^<f> d'4, d<t> 3^ dx^dic^ ' d-Xc^du^ ' dx^ii^ ' djc^ii ' Br,' 3iCjj a««^ av ^<f> 3'*^ d<f> Byfr ar>i' a^3.^' a«s3?*3 ' a^^jSw,' 3.3' dx^ d^4> d'if. 3*0 5^ d<f> d^lt dx^Ui ' dx^d^Uj ' dx^duj' dx^du^ ' dx,' ^x^ d<f> dtf, 9^2 ' Btf, 9«s' 3<^ 3«/ 0, 3«3' Bylr 0, (10) Digitized by Google PARTIAL DIFFERENTIAL EQUATIONS. I73 in the third the same expression with <f>, \p interchanged. The array is thus practi- cally unchanged by interchanging the sets x and w, as should be the case. § 24. This transformation may be accomplished by taking the equations from which may be deduced d_ I d^ du{\ __ ^ /^ 9<^ dUi dx^ \ i dui dxj dx^ \ i bui d(x^ ^^duiduj ^ff> duj d^<f} duj . d^ dui i j duidi(j dx^ dx^ i duidx-i dvc^ i duidy -^^ dx^ i duidz ^^ dx^ — ^^ ^^ ^M»rf^^j , ^ 9^<^ dUj d^4> duj 9^^ rfwi i j dui^Uj dx^ dx^ i duid.Vcj^ dx^ i duidy ^^ dx^ i dicidz ^^ dv^ ' Now Pi, q^y p^y q.^ are given by the relations and hence this equation may be wiitten * \dx^ d{x^,y,z) J i \dx^ d{x^,y,z) J (11); in this if), xff may be interchanged so as to give another equation. Now, suppose ^ = aj, X = % to be two of the four equations connecting ^1, ^^2j ^3? ^4 with ccj, x^y which yield a new complete primitive, and that y, z have been eliminated from ^, x by means of the equations <^ = 0, ^ = 0, then the deriva- *^^^® dx* dx' ^^'^ ^^ given by the following relations : — ^ 9<^ dUi _^ Q i diti dxi ' ^d±dui _ ^ i 9m; dxj .d0 dUi , 00 i oiii dx^ co\ i dui dj\ D./?! ' and similarly for the derivatives with respect to x^. Substituting the values hence found for these derivatives in the equation (11), we have an equation linear in the Jacobians of the form l^''y(i=l,2;J=l,2.S,4). Digitized by VjOOQ IC 174 MK. A. C. DIXON ON SIMULTANEOUS the coefficient of the Jacobian written being the minor of g^rg^ ^^ ^^^ determinant (10). Hence the constituents in the second column of the transformed array are as stated, and those of the third are found in like manner. It is not, of course, neces- sary that these columns should be the same as would be found by actual substitution of the values of pi, jo^, Ji, q^ in the columns of the original array ; a linear trans- formation is allowable, with constant or variable coefficients. The above process gives fifteen independent rows of the array ; the others are deduced from the consideration that y, z are known in terms of x^^ x^^ t/j, tfcg, u^^ u^ from the equations ^ = 0, ^ = 0. Examples. § 25. I. As a first example of the method of solution, take the equations ttj = ttg, ^1 = ^3, where aj, ^S^ denote known functions of ajj, jp^, q^ and ag, fi^ known functions of a^2> P2> ?2- In the array (5) multiply the seventh row by ^~^^y the fifteenth by g;^^' ^ L the twenty-fourth by - ^^^ ^l , and add. The result in the first column is d{aL^y fii), in the second, by virtue of the particular forms of/^ and/gj {^i.i^i} {?!> ?2} + K. 9i} {Q2rPi] + {Pi> ?i} {a^i» ^2} ^r 0, and in the third, i^i^Pi] {^2. 9^1} + {^1. qi) {PiyP2] - [Pu qi] {^1,^2} or 0. Hence a^, ^Sj are two functions satisfying the auxiliary equations, and a solution is given by finding p,, q,, p^, q^ from the equations ^1 = «2 = ^> ^1 = ^2 = &> and integrating. Two constants will be introduced by integration, so that the result is a complete primitive. § 26. II. Take, secondly, the equations y = Pi^i + F (0:2, i>i, g^i, i>o, q^\ z = q^x^ + G {x^, ^1, 5i, 7)2, q^y Here the twenty-fourth row is ^bi> ?])> 0, 0, Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 175 so that pi, 9i are two functions satisfying the auxiliary equations, and the integral is to be foimd by putting p^ = a, 5, = h. Thus we have y — axi = ¥ (Xz, a, h, p^, q^), z~bXi = G (x^, a, h, p^, q^), or 7i = F{la,h,v',i% C = G{i,a,b,y,',C), where ^ = cCg, 7^ = y — ax^^ ^= z — hx^^ r,' = d7,/d^, i' = dildl These are ordinary differential equations, the solution of which will involve two new arbitrary constants and so constitute a complete primitive of the original equations. § 27. HI. The equations y = l^i^i + P^^ + <^( Ply P2> ?i. <1^\ z = q^x^ + q^x^ + V^(^i, Pa, q^, q^\ are of special interest, because more complete primitives than one can be found. The obvious solution is p^ = a^, p^ = ag, qi = 61, q^ = &2> y = a^Xj + a^Xc^ + <^(ai, ag, &i, 62)^ z = b^x^ + h^x^ + y^{a^, %, 61, h^. Suppose a^ a^^ h^y h^ to be variable, but functions of one variable only— say aj, then their variations must satisfy the relations x^ da^ + x^ da^ + ^<^ = 0, Xy dh^ + x^ d\ + ^'A = 0. These define x^, x^, and, therefore, also y, 2 as functions of a^, unless the determi- nants of the matrix da^y da^y d(f> 11 db^y db^y d^ I vanish ; it is necessary, then, that these determinants should vanish. Thus ^u &i» ^2» ^2 ^re connected by two ordinary differential equations. We may assume any third relation connecting them at will ; suppose b^ = ^{^1)9 ^ denoting an arbitrary function. Then by integration we may suppose a^, &2 fo^^d in terms of a^. Also a^ is connected with x^y x^ by the relation ^^ + ^2 da, + da, "" "* SO that ttj, 61, a2, 62 ^^e all known in terms of Xj, oTo, and by substitution the values of v, z are found. Digitized by VnOOQ iC so that 176 MR. A. C. DIXON ON SIMULTANEOUS § 28. The solution may be verified. We have taken h^ = F(ai), a known but arbitrary function of a^, and a^, h^ other functions of aj, such that Then we have the further relations which are of course not distinct. Also P^-^^^ de, [^ ^^ ^2 rfoj ^^ 3a, ^^ 3^3^01 ^ 36i da^ ^ 36, (iaj " ^^' and in like manner p^ = a^. Again 2? = 6i iTi + &2 ^2 + ^{<^iy %» ^i» ^2)^ and ^ «- 7, I ^ L ^*l I ^ rf^g 3^^ 32fr da2 3i/^ 9^1 - oi i- ^^ i^a^i ^^^ "<■ ^2 ^^ -h 3^^ i- 9^^ ^^^ "^ 36^ rfo, "^ (^6. daj "" ""^^ and similarly g^ = 6^. Hence the original differential equations are actually satisfied. If the arbitrary relation assumed — which may if convenient involve more than two of the parameters — contains two arbitrary constants, the new solution will generally be a complete primitive, since two more constants are introduced by integration. *t * The ordinary equations to be integrated may have a singular solution with one arbitrary constant, or with none : if the arbitrary function has been chosen so as to involve three or four arbitrary constants, the whole number being thus raised to four, the solutipn so given may quite well be a complete primitive, and, in general, will be so. t The above investigation in a modified form shows how to find integrals of a system of three equations /2 (w, J, i?i, ;?2, gi, ?2) = 0, L (12) /s (w, t;, i>i, i>2, gi, ^2) = o,J where u = piXi + p^x^ - y,v = qiXi + q^x^ - z. One solution is to take t^)?M'i,^2)^ii $2 £U3 constants connected by the three relations (12) ; if they are not constants we have du =■ Xi dpi + iCa dp^^ dv = xi dqi + X2 dq2. Digitized by VnOOQ iC PARTIAL DIFFEllENTIAL EQUATIONS. 177 § 29. Let us now consider the new solutions of the auxiliary equations, given by the new complete primitive. The old solutions are the six pairs of the form a;,-, xj and the six of the form t<,-, tij, where Mj = pi, % = p,, Wj = g-j, tt^ = q^. The bi- differentials of these twelve satisfy the relation d{y, z) -pAx^, z) -pod{x^, z) - q^d^, x^) - q^y, x^) + {p^q^ - Piqi)d{x^,Xi) + x^ _^3^ 3f «1 3* 3j, 3^ «2 3<^ 3?8 3^ 3^ ^'(Pi.l'a) + <^(Pi. 92) + «?(;>2. g'2) + X, .3* 3<^ 3f «1 3^ 3<^ 33i 3^^ 3^ + 3j. 3« 3(^ a?.' ^ Xl 3Vr ^(Pi> 9i) ^(i^2> 9l) <^(<?l» ^2) In the auxiliary equations we may take ajj, x^^ pj, p^^ q^y q^ as independent variables, since y, « are given explicitly in terms of these six. From (12) follow three more relations connecting the six differentials du, dv, dpi, dp2, dqi, dq2, so that their ratios are determinate, and therefore w, r,^i, qi,p2, Ja can only be functions of one variable. The two equations last written will then, generally, give xi, x^ in terms of this variable, which may not be. Hence we must have du = Xdv^dpi s= \dq\^dp2 = ^^2j and since df\ = 0, c(^2 = 0, dfz = 0, and du^ dvy dpi, dqi, dp^, dq^ do not vanish, X must satisfy the equation : Sib dv ' dpi dqi ' dp2 dq2 \^ + ifi X^ + ^ \^'f^+^l du dv ' dpi dqi ' dp2 dq2 du dv ' dpi dqi ' dp2 dq2 : If X satisfies this equation the differential relations du = \dv, dpi = A^i, dp2 - Mq2 reduce to two only, since u, v, pi, ^i, p2, ^2 are connected by the equations /l = 0,/2 = 0,/5- 0. By integrating these two we find two more relations involving two arbitrary constants. Hence, wq ^niay suppose V, pi, p2y qi, 92 expressed in tenjils of t/, atid fin4 a solution by eliminating u from the following : — ' , u :^ piXi +p2X^ - y, V = ftJCl + 22»2 - ^, 1 =. Xi dpifdu + ^n dp2ldu, VOL, CXCV. — ^A, 2 A Digitized by VnOOQ iC 178 . MR. A. C. DIXON ON SIMULTANEOUS Then rf(«i» y) = pA^i> ^2) + [^x + ^J^i^uPi) + (ajj + ^J%i» P2) + g^ <H^u qi) + and similar expressions may be found for (/(ajgj y), c?(a;j, 2), ^(arg, 2) in terms of the bidifferentials of the pairs of independent variables. Let c„ ^2, C3, c^ be the constants of integration in a new complete primitive found by the method of §§ 27-8. Let X be the common value of the ratios dp^Jdqi^ dpjdq^, d^/d^. Then, after integrating the equations dpjdq^^ = dpjdq^ = d^/d^ (= X) by help of an assumed relation connecting, say, jp^, g^, jpg, g^, (?i, Cj we have four relations among and we may therefore suppose pi, pg* S'l* 5^2 expressed in terms of X, Cj, c^, C3, C4, unless X is a constant, and therefore itself a ftmdtion of Cj, Cj, C3, C4. Then dpi -^ X d^Tj, dp2 — X dq^^ d<ff '^ Xd^jt will be linear combinations of dcj, rfcg, cfcj, 0^4, and so will some such expression as adpi + $d\^ where a vanishes if X is one of the constants or a function of them. Conversely, rfcj, rfcg, dcg, dc4 will be linear combinations of dpy — X d^i, c?p2 " ^ ^9^2> ^ — ^ ^^> a ^JPi + /8 ^X, and the bidifferentials of Ci, e,, Cg, C4 in pairs will be linear combinations of the six following expressions : — ^ {Pi> P2) -^ >^d{qv p%) - ^{Pi^ q^) + ^^^ (?i> qz)^ d (i>i, <^) - Xd (^1, <^) - Xd ( Pi, i/r) + fc«d (/?!, 1/1), ^ (i>2> <^) - ^d (^2, <^) - \d {p^, i/r) + X^d (92, i/r), /8d (jpi, X) - akd {q,, p,) - fiXd (q,, X), a^ (i>2> Pi) + fid ( P2» ^) - a^^ (?2» Pi) - fi^d (^2, X), acZ(<^, p,) + i8d(<^, X) - aXdii^, p,) - 0XiJ(f, X). These are combinations of the bidifferentials of p^, p^^ q^^ q^y in pairs, with the expremonfl d{<^,X)-Xci{iA,X). Digitized by Google • PARTIAL WFPEBENTIAL EQUATIONS. 179 Now X is a definite function of ojj, aug, pi, p^, qi, q^, given by eliminating the differentials from the equations dpi s= \dq\> dpi ^ A<ig2, d^ = )di^. By means of the first two, the third becomes and the fourth The result of ^^lioMiifiAion i« th^rofoise This shows the form of X as a function of asj, jCj, ^Ji, jjj, ^j, ^j, not involving Cj, Ca, Cj, C4. Now this choice of X makes it possible to choose coefficients A, B, C, E, F, G, such that ^1 ^i + ^» dPi -\- d<f> = A {dpi — Xdg,) + B {dp.^ — Xdgj) + C {d^ — Xdi|»), Xi dqi+x^dq^ + d^li^E {dpi — kdqi) -f F (rfpa - Xtijj) + G (rf^ - Xdt^). Thus Hd{Pi, X) - \d{qi, X)} +B {rf(i>2. X) - Xd{q„ X)} + C{d(<^, X) - Xdi^j., X)} = xid{p„ X) + a;ad(pj, X) + d{(f>, X) = multiples of bidiffereatials of jp^, p^, qi, q^ + ^{«i <^ bi. ari) + Xid (pi, x^) + rf (^, a;,)} iW r * I + a^{»i <^ 0>;, .aJ«) + XafiiPp x^ + d {^, aa)| ~ a^l*^ ^^' ^») ~ P^^ ^^^' ^»q "*" ^l*^ ^^' ^^^ ~ ^^'^ ^^^' ^*q + miiltiples of bidifferentials of ^], p^, q^, q^. 2 A 2 Digitized by VjOOQ IC 180 MR. A. C. DIXON ON SIMULTANEOUS In like manner E {d(i)„ X) - Xd{q„ X)} + F {d{p,, X) - \d{q,, X)} + G {d{<f>^\) - \d (i|r, X)} = ^ {d {z, x^) - g.,d (a;^, x,)} + ^-{d{z,x^) - q,d {x,, x^)} + multiples of bidifferentials of p^, p^, g^, ^g. Hence the three expressions d{p,,\)^\d{q,,\), are all reduced to the same, save for a factor, by adding or subtracting multiples of the bidifferentials of Xj, a?^, ojj, x^ and of u^y %, 1^3, t^^ ; the same is therefore true of the bidifferentials of Cj, Cg, C3, c^. Hence all the new complete primitives found by the method of §§ 27-8 only add one to the eleven known " bifimctionally " indepen- dent pairs of functions satisfying the auxiliary equations ; one more pair, leading to a fresh complete primitive, is yet to be found. § 30. These results may be used to construct examples of bifunctions. For instance, the equations % = q^x^ + q^^ + p^y lead to the following case among others : — In the equations ^ = ^ = -^^^ = X dq^ dq^ dp^ put (/o = X + a, dq^ = rfX, and integrate. Thus 2^2 = X* + 6, 351 = X' + c, 4pi = X* + e, and the arbitrary constants a, 6, c, e in the new solution are respectively equal to q,2 — X, 2^2 — X^, Sq^ — X^, 4pi — X*, where X^ajj + Xg + X = 0. Now from § 29 it follows that d (c, e) can be expressed in terms of d (a, 6), the bidifferentials of x^, x^, y, z and those of p^, p^y ^u 9^- For convenience, let us write w, V, w, Xy y, z for Xj, X, py p^y qi, q^ respectively ; then for x^ we must put — v(l + wv), for y „ „ tw — xv{l + uv) + y, for z „ „ yu — zv(\ + wv) + x, Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 181 so that the eight original variables connected by two equations are now expressed in terms of six. Thus d (3y — t;^, 4ti; — v*) can be expressed in terms of d (2; — y, 2a? — v^), the six bifferentials of Wy a:, y, z and those of u, — t;(l + uv\ wu -— xv{l + uv) + y, yu ^ zv{l + uv) + x, that is, of u, V, wu — xv{l + t^v) + y, yte — zv{\ + wv) + x. There is no difficulty in finding the relation. It is M»(i(3y - t;», 4i^ - t;4) - 62;2(1 -f uvfd{z - i;, 2x - v^) - 12w2rf(y, w) + 12v2(l + wi;)»d(z, x) + 12i;*{(l + ^v) {y — 2;t;*) ^ ^(^ — ^)}d{Vy u) — I2i;*{l + uv)d{Vy yU'^zv '^ uz't? + x) + 12i^t;*c?(v, t(n^ — ajv — t^on;^ -f y) = 0. Here then we have an identical linear relation connecting the bidifferentials of seven pairs of fiinctions of six variables. Any one of the seven pairs is accordingly by definition a bifimction of the other six. Second Application. § 31. Take now a differential equation of the second order, where p, q are the first and r, 5, t the second partial derivatives of z with respect to Xy y. A complete primitive will consist of a single equation in aj, y, z involving five arbitrary constants, say aj, ag, ^3, a^ a^. If we form the first and second derivatives of this equation we shall have, in all, six equations from which a^, aj, aj, a^ a^ can be foimd in terms of a:, y, 2;, jp, q^ ?% 5, ty and the original differential equation will be the result of eliminating a^, a^, ag, a^y a^. Let w^, w^, Ug, -w^, Wg represent the expressions found for a^y a^y %, a^, % respectively, in terms of Xy y, «, p, y, r, 5, t. Then from the equations /=0, Wi=ai, Wgsa^, by differentiating, we can form six equations which will involve the third derivatives of 2 ; by eliminating these we deduce the following two differential equations to be satisfied by Wj, u^ : — Digitized by VnOOQ iC 182 MR. A. C. DIXON ON SIMULTANEOUS J{x, r, t) + pJ{z, r, ^ + rJ(jp, r, «) + s3{q, r, ^ + J(y, «,-<) + g^^ «, e) J(a?, «, r) + pJ(2, 5, r) + rJ{p, «, r) + 8J{q, Sy r) + %, i, r) + 5J(2:, ^, r) + sj{p, i{, r) +tJ{q,t,r) = 0. Here J ( ) denotes the Jacobian of /, u^, Uc^ with respect to the variables specified. These equations express the conditions which are necessary and sufficient in order that dz = pdx -f qdyy dp = rdx + sdy, dq 9 3ch + idy may be integrable without restriction, when r, 5, ^ are given in terms of Xy y, 2;, |?, g', by the equations the conditions must of course be satisfied by any three of the six functions Wj, Wg, tig, %, tig, yi We thus have forty equations, of which only eight can be algebraically independent I '32. The conditions to be satisfied by ti^, u^ are linear and homogeneous in their Jacobians with respect to the eight variables x, y, 2J,p, g, r^ s, t; of these, one is given in terms of the rest by the equation /= 0, and may, if convenient, be sup- posed not to occur in u^^ u^ : hence the auxiliary equations in this case have seven independent variables and the dependent variables do not occur explicitly: to find a solution we are thme&r^ to Soim a OQmpkte bidi£S^WPtial, whioh ^all bp a linear combination of the determinants of the following array : — d{r,s), -X-^Z-rP- -sQ d{r,ff, Ti + pZ + rf + iQ, _ Y - ^ - ^P - -tQ d{s,t), Y + qZ + sP-\- tQ, • (^P.^, rT, -rS-AT (5) d{p,s). *T, rJR dip.,tl - rB - «S sB. «?(2. r). sT, -S-tT d{q,s), tT, sR d{q, t), - sR - <S, tR (10) d{z,r), i>T, -pS-qT d{z,s). ?T, pB. #,<9. -/)R-^, ^R d(x, r), T. -S d{x, s). 0, R Digitized by VjOOQ IC PARTIAL DIFFERENTIAL EQUATIONS. 183 (15) 4x,<), -R ^.r). 0, -T %.*)» T, %.o. -s. R Ax,p), 0. (20) d{9,pl 0, <H?^>pl 0. 0- rf(x,Qr), 0, %.^). 0, d{z,q), 0, (25) d(p,g), 0. ci(x,«), 0. %.^). 0, d{x,y\ 0, X,P. . . are written for df/dx, ,^fl^ . . . Of these twenty-eight rows, only twenty-one are independent* Fw instanoe, multiply the Ist, 2nd, 4th, 7th, 10th, 13th, 16th by - S, - T, P, Q, Z, X, Y respec- tively and add ; the resulting row is d(/r), 0, 0, which vanishes since f=Ohy hypothesis. Suppose d {uyy u^) to be the complete bidifferential formed from the determinants of the array, then to complete the solution we have to find r, 5, t from the equations and integrate the equations dz = pdx + qdy, dp = rdx + sdy, dq = sdx + tdy. It will amount to the same thing if we treat t*i as known in the auxiliary equations. They must be satisfied if Wg, u^, u^ are substituted in turn for tig. Now two homogeneous linear partial differential equations in seven independent variables can at most have five common solutions, and here one of these, tt^, is known ; the other four may be taken as u^, t^, ^4, u^. § 33. Any two of the five ftmctions x, y, 2, p, q will satisfy the auxiliary equations, but as we have to solve for r, 5, ^, these solutions will not serve our purpose. They are ten in number, and ten more will be given by taking in pairs the expressions ti^, ^2> ^'3> ^4> '^5 given by any complete primitive. These twenty are not all bifimc- tiotially independent, for since there are three relations* among the ten expressions a?, y, 2J, p, q, Wi, 1/2, W3, «4, z^6, * Compare § 34> {k 184. Digitized by VnOOQ iC 184 MR A. C. DIXON ON SIMULTANEOUS three linear relations can be formed connecting the twenty bidifferentials ; one is formed from each pair of equations as at § 17 (8). Hence seventeen biftmctionally independent solutions of the auxiliary equations are known when we have one com- plete primitive. The ftdl number is nineteen (t^ " 2 j, and in order to know all we must have one, or possibly two (see § 41, p. 190), more complete primitives. § 34. New solutions found by varying the parameters may be divided into two classes, according as the parameters are or are not all fiinctions of one variable ; solutions of the former class only occur in exceptional cases, and the principles of § 21 apply to them with slight modification. Let the three equations connecting «i Vf 2J, p, q, tti, Wg, ^8. ^4* ^6 be <i>t{x, y, z, p, q, u^, u^, Wj, W4, u^) = (t = 1, 2, 3) ; (the forms <^|, <^2, ^g are not unrestricted, but must be such that the following rela- tions hold identically or we may take <^i as not involving />, q and ^^^^ P d<f>Jdz + 9<^,/3ic then the variations of the parameters must satisfy the three equations [l^^d^r ^ {i = 1, 2, S), in order that the same relations may subsist among a?, y, 2, p, q, r, s, t and the para- meters, as held when the parameters were constant. If the parameters are functions of one variable, their forms must be so chosen that the three equations last written reduce to one only, otherwise we shall have five relations connecting x, y, z, p, q with this single variable. § 35. If the parameters are not functions of one variable, only the equations are equivalent to six, and determine the partial derivatives of v^, u^ u^ with respect to Wj, % in terms of the five parameters and aj, y, 2;, jp,, q. By help of the relations ^i = we may suppose x, y, 2, p, q eliminated and thus arrive at a system of four partial differential equations connecting Wj, u^y w,, w^, u^. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 185 The original system may also be taken to consist of four equations connecting five variables a:, y, z, p, 9, namely : dzjdx = p, dzldy = 5, dp/dy = dq/dx f{^y y. 2J, p, g, dp/dx, dp/dy, dq/dy) = 0, and so the method of variation of parameters does not lead to any simplification of the problem in general. § 36. The interchange of variables and parameters is again possible ; it is, perhaps, made clearer by taking three equations of perfectly general form, <f>i (o^i, x^y x^y x^, x^y u^y u^ u^, u^, Ug) = (^ = 1 , 2 , 3), connecting two sets, each of five quantities. Whichever set we suppose constant and eliminated by diflferentiation, we are led to a system of four partial diflTerential equations connecting the quantities of the other set, two of the five being taken as independent variables. A new solution of either of these systems of differential equations will in general yield a new solution of the other. Suppose, for instance, that we have a new solution of the u equations ; this gives t^3, u^ Wg, say, in terms of f.^, u^. Then the six equations included in 't^^' dur- 0(i= 1, 2, 3) give two relations among cc^, . . . Wj, t^^, since the four differential equations, which are consequences of these six, are supposed satisfied ; by the help of these two, u^y u^y may be eliminated from the three relations <^i = 0, ^^ = 0> ^3 == ^> ^^^ thus three relations are given connecting aj^, iCg, x^y x^y x^ ; these three will constitute a solution of the x system of differential equations. § 37. In this more general case there will not seemingly, as a rule, be any more solutions for either system of differential equations. For the derivatives, say, of ^> ^4> ^5 with respect to x^y x^ are given in terihs of these five variables and two others, say Mj, u^. The forms we may assign to u^y u^ are then restricted by three differential equations derived fi'om the three conditions -i^ J^'^L^ /'r — 3 4 ^) dx^ cLc^ ■" cU^ clx, ^ "" ""' ^' ""'' and thus, generally speaking, no forms of W|, iig will be suitable. In some cases the conditions are not inconsistent, and we may form an array by the method of § 11 such that if d{0yx) is a combination of its determinants, then ^ = a, x = &> ^1 = 0. VOL. CXCV.— A. 2 B Digitized by VnOOQ iC 186 MR. A. C. DIXON ON SIMULTANEOUS <^2 = 0, ^3 = Mrill give suitable values for Wi, Wj. This array will have four columns and forty-five rows, ten such as d{xi,xj), 0,0,0, ten such as d (m„ uJ), 0, 0, 0, and twenty-five of the following type. ■ In the first column there is d{xi, Uj), in the (r + l)th the minor of S^<l»r/dx,duj, in the determinant d^<f>r a^r 3'^r ^<l>r 3*^r d.i\ dtii 3^3 dii^' Sij 3mj' B'j 9«j' 8/j 3j<j 3'4»r 3dii 3r^ 3^s 3^1 3^ 3*3 3^8 3^ 3«8 3^ 3^3 3*3 3tB, for 3^ 3^, 3^3 3c^ r = 1, 2, 3, 3^<^, a^j 3»<j 3^ 3<ib 308 3^5 3<j^ 3^ 3^8 3*1 3z^3 3ii 3^4 3^ 3l^6 3*2 3*3 9mj 3?^i 3*3 3^ 3*2 3*3 3ws 3i^3 3*2 3(^g 3*2 3^ 3^6 3i^6 This interchange of variables and parameters may take place whenever their numbers are equal, the diflferential equations being of the first degree. Eooamjyles. § 38. L As an example of the method of solution take the equation a = /8, where a is a function of r, 5, p — sy^ x and fi a function of s, t^ q ^ sx, y. In the array (§ 32) multiply the first row by 9a/9r, the fifth by 9a/9jt>, the fourteenth by 9a/9a:, the seventeenth by — 5 9a/9p, and add ; the resulting row is d{a, s), 0, 0. Hence we take a = /8 = a, 5 = 6, z = hxy + X + Y, X being a function of x only and Y a function of y only. Then a = a is a relation connecting x, dX/dx, d^X/dx^, and fi = a is a relation connecting y, clY/dy, d^Y/dy^^ and by solving these for X, Y respectively we shall have the complete primitive. Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 187 § 39. IL As a second example take the equation F(r, s,t,p — ^,q — ty,z — qy + \tif^, or) = 0. Here the third row of the array is d{s, t\ 0, 0, so that the fiinctions «, t satisfy the auxiliary equations. Put, then, 5 = a, t = 6 ; thus gr = owr + 6y + c 2 = aa?y + i 5y» + cy + X, the last term heing a function of x only. The differential equation thus becomes F{d^X/clx\ a, 6, dX/dx, ax + c, X, x) = 0, an ordinary equation of the second order giving X in terms of x and two more arbitrary constants ; hence the finding of a complete primitive is reduced to the solu- tion of the equation last written, § 40. ni. If the equation is of the particular form F(r, s, t,p ^rx ^sy,q — sx^ ty, 2: — px — qy+^rx^+sxy+^ty^) = 0, the first three rows of the array are d{r,s) d{r,t) d{s, t) 0. Hence any two of the three functions r, s, t will satisfy the auxiliary equations, and a complete primitive is given by putting r ^= a^ s = h^ t = b. Hence p = ax+hy'\-g, q = ^+6y+/ z = c + gx +Jy + ^{aa^ + 2hxy -f bf), where a, 6, c,/, g^ h are constants satisfying the relation This is a case in which other solutions are readily given by supposing the para- meters variable and functions of one variable only, say a. The variations must satisfy the conditions x^da + 2xydh + yHh + 2xdg + 2ydf+ 2dc = 0, a: c?a + y rfA + dgr = 0, « dA + y d6 + cZ/ = 0, 2 B 2 Digitized by VnOOQ iC 188 ME. A. C. DIXON ON STMITLTANEOUS whence follows xdg'\-ydf-\-2dc — 0, a simpler relation that may be taken instead of the first of the three. These equations will define x, y in terms of the single variable a, unless all the first minors of vanish. dkt dh dg dh dh df dg df 'idc We thus have three ordinary differential equations connecting a, 6, c, f^ gyh\ they are connected also by the relation F (a, ^, 6, g, /, c) = 0, and the fifth relation among them may be chosen arbitrarily, so that we may put h = (j) (a), an arbitrary function. Then we have db/da = {f (a)}^ df/da = f (a) dg/da, 2dc/da = {dg/daf, F{a,<f>{a),b,g,f,c) = as the equations determining 6, g^ /, c in terms of a. These are to be integrated, and then a is to be eliminated from the equations 03 + y dhjda + dg/da = 0, z = c + gx +fy + ^{ax^ + 2hxy + hy^). The result of elimination will be a solution of the differential equation. Three constants are introduced by integration, and thus, if the function ^ involves two constants, the new solution will generally be a complete primitive. § 41. The new complete primitive gives new solutions of the auxiliary equations which we shaU now examine. Let a^, ag, Og, a^, a^ be the new set of parameters. Then a, h, g, 6, c, / are connected with these parameters by five equations, one of which is the original equation F = 0. These five relations are such, that if dh = Xrfa, dg = /jida , then db = \^da^ 2dc = /ji^cZa, df = Xfxda ; of these five, the first two define X, fx in terms of a, a^, a^, ag, a^, ag, and the others must then follow from the five equations that give A, g, 6, c, / in terms of a and the same new constants. Thus, in general, we may suppose a, A, g^ 6, c, f, /ji, expressed in terms of X, a^, a^, ag, a^, a- and the expressions will be such that dh — \da^ dg — [xda, db — \^da, 2dc — fjL^da, df — Xfida Digitized by VnOOQ iC PARTIAL DIFFEREXTIAL KQrATIOSa 189 involve only the differentials of «!, a^ fc^, •♦* *5- ^^ ^f these five is expites^hle in terms of the other four, since cF cF ?F rF rF rF da * ck ' c* o/ ^ </^ • flr while one of the relations connecting X, ft, a, 6, ... is aF , ^ar , pf , ^.?f , ^ aF , , ^?f Some expression such as vd\ — pda will also involve the differentials of a^, 04, a^, a^, ttj only. Hence the differentials of aj, a^, o^, a^ a^ will be linear combinations ot vdX — prfa, dh — Xc/a, rf^ — firfa, c/6 — X-t/a, rf/*— X/jw/a, 2dc — fiVa, of which the last five satisfy a linear relation. Thus the bidifferentials of a^, a^, a,, a^, a^ in pairs will be linear combinations of the bidifferentials of a, 6, c, /, gr, A (only five of the six need be used) in pairs^ and of the expressions d{h, X) - Xrf(a, X), %, X) - ^d(a, X), c/(5, X) - X«rf(a, X), d{f. X) - X^rf(a, X), 2d(o, X) - /^^^(a, X), of which last five, only four are independent. Now X, II are connected not only by the equation dF dF dF dF dF QF ^ + ^a^+ '*^+ ^V+ ^37 + ^/^V = ^' but also by the equation 35 + Xy 4- /t = 0, so that they are definite functions of x, y, a, h, /, g, h. Again p = ax + hy + g, d{p, x) — hd{y, x) = xd{a, x) + yd{h, x) + d{g, «), «^(p. y) — «<^(». y) = ^<^(«. y) + W(^. y) + %. y)- Thus y{<f(A, X) - Xrf(a, X)} + {%, X) - ftd{a, X)} = xd{a, X) + yd(A, X) + %, X) = a^ [^(p. ^) - ^«'(y. a')] + aj; I^^p^ v) - ^K^> y)l + multiples of bidifierentials of «, 6, c, /, g', h. Digitized by VjOOQ IC 190 MR. A. C. DIXON ON SIMULTANEOUS In like manner y{d{h, X) - X^a, X)} + {d{f, X) - \iid{a, X)} + x{d(h, X) - \d{a, X)j = xd{h, X) + yd{h X) + d(y, X) = a^ \d{<i^ ^) - &%. «^)] + ^ \A{^^ y) - ^d{x, y)\ + multiples of bidifferentials of a, 6, c, yj gr, h. Lastly, 2xy{d(h, X) - Xrf(a, X)} + y*{(^(6, X) - \H{a, X)} + 2ir{%, X) - ixd{a, X)} + 2y{d(/, X) - X,irf(a, X)} + {2ti(c, X) - /.V(a, X)} = a?d{a, X) + 2xyd{h, X) + y^d(6, X) + 2xd{g, X) + 2t/d(/, X) + 2rf(c, X) = 2{d{z, x) — {hx + by +f)d{y, x)}d\/dx + 2{d{z, y) - {ax + % + 5r)d(^, y)} d\/dy, + multiples of bidifferentials of a, 6, c, /, g^ h. Hence, in all, nine combinations of the ten bidifferentials of a^, ag, ag, a^, a- can be expressed in terms of the bidifferentials of x, y, z, p^ q and of a, 6, c, fy g,h\ that is, in terms of the bidifferentials of the seventeen known independent pairs of functions satisfying the auxiliary equations : thus the new complete primitive adds only one to the number of these known bifunctionally independent pairs, and one more must be added in order to give the full number. This theory enables us again to construct examples of bifunctions of a number of known pairs which may reach eighteen. § 42. The foregoing investigation may be modified so as to give singular solutions of a pair of differential equations of the form in question, say Fi(r, s, t, p, q, z) = 0, F^Cn s, t, p, q, z) = 0, where p = p -- rx ^ sy, q = q — sx --ty, A complete primitive would be given by supposing r, 5, t, p, q, z constants con- nected by the above equations. Another solution would be given by solving the total differential equations found by supposing the relations Digitized by VnOOQ iC PARTIAL DIFFERENTIAL EQUATIONS. 191 xdr -{-y ds + dp = 0, xds '\' y dt -^ dq = 0, xdp+ y dq + 2dz = 0, to reduce to the same relation linear in x and y. That is, we must solve the system ds = \dr, dt = X^dvy dp = fidr, where X, /x are given in terms of ^, q^ 2, r, s, t by the relations 9Fi 9Fi vaSFj 9Fi 9Fi ^ 1 o9Fi ar + V + ^V + ^r + ^'^ar + ^'^ar =^' 9Fg 9F3 xo9F 9F3 9F3 - 2 9F, and 5^, 2J in terms of jp, r, 5, i by the relations F^ = 0, F2 =0. The complete primitive of these ordinary equations will involve three arbitrary constants, and there may be singular solutions with a lower number ; none of these will therefore constitute a complete primitive of the partial diiSferential system Fi = 0, F3 = 0. Digitized by VnOOQ iC Digitized by Google [ 193 ] V. The Velocity of the loris ^produced in Gases by Rmtgen Rays. By John Zeleny, B.Sc., B.A., Assistant Professor of Physics, University of Minnesota. Communicated by Professor J. J. Thomson, F.R.S. Received February 5, — Read March 1, 1900. § 1. Introduction. The electrical conductivity which is imparted to gases by their exposure to Rontgen rays has been explained by J. J. Thomson and E. Rutherford* on the hypothesis of a formation of oppositely charged carriers throughout the volume of the gas. The motion of these carriers or ions when in an electric field constitutes the observed conductivity, and the recovery of the insulating property of a gas after an exposure to the rays is due partly to the recombination of the oppositely charged ions and partly to their impact with the boundaries. An estimate of the sum of the velocities with which the positive and negative ions move in air when in a unit electric field was first obtained by J. J. Thomson and E. Rutherford, and later E. Rutherford,! by the same indirect method, determined the sum of the velocities of the ions in a number of gases. This method involved the determination of the rate of recombination of the ions, the saturation current obtained through the gas by the use of a strong electric field, and the current obtained with some small non-saturating electric force. E. Rutherford also describes an experiment in which the velocities of the two ions in air were obtained separately by a direct method, and found to be approximately equal. The writerj has since shown that in general the two velocities are not equal, and for those gases for which the ratio of the two velocities was determined the negative ion moved the faster in nearly all cases. The values of the velocities of the ions have recently been applied by J. J. Thomson§ and J. S. Townsend|| in the determination of important physical quantities, and it seemed desirable that a redetermination of the values of the velocities be * J. J. Thomson, and E. Rutherford, * Phil. Mag.,' November, 1896 t E. Rutherford, * Phil. Mag.,' November, 1897 X J. Zeleny, * Phil. Mag.,' July, 1898. § J. J. Thomson, *Phil. Mag.,' December, 1898. II J. S. Townsend, *Phil. Trans.,' A, vol. 193, 1899. VOL. cxcv— A 266. 2c 9.11.1900. Digitized by Google 194 MR. J. ZELENY ON THE VELOCITY OP THE IONS undertaken, partly because of advances in our understanding of some of the intrica- cies of the conduction, and partly because it seemed desirable that a satisfactory direct method be devised whereby the velocities of the two ions could be determined separately, and in which the experimental conditions could be subjected to a number of variations sufficient to ensure freedom from serious errors. In undertaking this, an attempt was first made to use a modification of the method employed by the writer in the determination of the ratio of the two ionic velocities, which is described in a previous paper. The ions were made to go against a stream of gas in a tube by means of an electric field, and their velocity was compared to that of the gas stream. The presence of the gauzes necessary for the production of the electric field was found, however, to disturb the gas stream sufficiently to produce a turbulent motion in it and so prevented the attainment of absolute results. The method which was then developed, and the one with which all of the results of this paper were obtained, also consisted in directly comparing the ionic velocity with that of a stream of gas, but avoided the difficulty of the above by having the electric field at right angles to the gas stream. § 2. The Method Used for Determining the Velocity. A stream of gas is passed between two concentric cylinders which are kept at difierent potentials, and which at one place are traversed by a beam of Rontgen rays. The ions which are produced between the two cylinders by the rays are carried along by the stream of gas and at the same time, under the influence of the electric force, they move at right angles to the axis of the tubes. The resultant paths of the ions are inclined by an amount depending upon the relative value of the velocity of the gas stream to that of the ions. Let CC in fig. 1 represent a section of a portion of the outer cylinder, and DB that of the inner one, and let dd represent a narrow beam of rays traversing the two cylinders at right angles to their common axis. When the two cylinders are at Fig. 1. Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 195 different potentials and the gas between them is at rest, an ion starting from the point d at the inner surface of the outer cylinder will move directly across to / imder the electric force. But when a stream of gas is passing between the cylinders from right to left, the ion will also be carried along by the stream, and so follow a path somewhat like that represented by the curve dk^ finally reaching the inner cylinder at some point, i, which can be determined. The paths of the ions are not straight lines, because the electric intensity and the velocity of the gas stream vary from point to point between the cylinders and according to a different law for each. The distance X that the ions have been carried along the tube by the gas stream while they are crossing between the two cylinders under the electric force is a measure of the relative velocities of the gas and of the ions, and so may be used in determining the velocity with which the ions move in a given. electric field. Let the outer cylinder be kept at a potential of A volts and the inner one at zero potential. Let h be the inner radius of the outer cylinder and a the outer radius of the inner cylinder. Then the potential at any point between the cylinders at a distance r from the common axis of the two cylinders is and the electric intensity at this point is dR A dr rlogebfa ' V /• If we let V represent the velocity with which an ion moves when in an electric field whose intensity is 1 volt per centim., and assume that its velocity is proportional to the strength of the field, then at a point whose electric intensity is represented bv equation (2), the radial velocity of the ion will be V = -^ . (3). The ion being carried by the moving gas also has a motion along the tubes. The velocity of the gas stream at any point depends upon its distance from the axis of the cylinders, which will be called the x axis. Suppose that at the distance r from this axis the gas velocity is u. The motion of the ion is represented by dx u ~dT~ Y ' ' ' \ (^/' and substituting the value of V from (3), 2 c 2 Digitized by Google 196 MR. J. ZELENY ON THE VELOCITY OF THE IONS The distance X travelled by the ion in the direction of the x axis while it is traversing the whole distance between the cylinders, i.e., from r = 6 to r = a, is ^='^:'"f/-^'- («)• Now the average velocity of the gas stream as measured by the quotient of the total volume of gas emitted in a second by the area of the cross section is From (6) and (7) ^= 2kv ^^»a • • • (^)' ^""^ ^= 2AX ^^g-a " * ' ^^^^ This gives the value of the ionic velocity in a unit field in terms of quantities which can be experimentally determined. The time required for the ions to pass from one cylinder to the other is The equations above apply to ions starting from the inner surface of the outer cylinder and moving inward to the inner cylinder. In practice it is not possible to limit the production of the ions by the Rontgen rays to the inner surface of the outer cylinder, so a narrow beam of rays is passed at right angles through the cylinders, as is represented by dd of fig. 1. Of the ions of this layer which move inward under the influence of the electric force, those that start from the circumference at d are carried the farthest by the gas stream before they reach the inner cylinder. Under these conditions the equations obtained can be applied by determining the point along the inner cylinder farthest from the beam of rays that is still reached by ions. For obtaining this point, the inner cylinder DB is divided at k into two parts, insulated from each other, the part B to the right being connected to earth, while the part D, to the left of the division at i, is connected to a pair of the quadrants of an electrometer. If a definite stream of gas is maintained between the two cylinders, then while the potential of the outer tube CC is above a certain value, all of the ions from the volume dd which move inward will reach DB to the right of the juncture k, and so the electrometer reading will not change. By gradually diminishing the potential of CC a value is finally reached such that the ions starting from the outer edge d reach DB just to the left of k, as wiU be indicated by a changing electrometer reading. The value of the voltage A in equation (9) is thus determined, and the value of X, which corresponds to it, is the distance from the beam of rays to the juncture k. In getting X the corrections which must be made for the width of the beam of the rays and for Digitized by VnOOQ iC PEODUCED IN GASES BY RONTGEN RAYS. 197 the width of the juncture k will be considered later. The apparatus as used will now be described. § 3. The Apparatus. The main parts of the apparatus are represented in fig. 2, where the lower part of the figure is a vertical section, while the electrical connections in the upper part are viewed firom above. Fig. 2. The outer cylinder, A A', had an internal diameter of 5*11 centims., and a total length of 142 centims. For convenience the length is shortened in the figure by the omission of two sections. The part to the left of V, 41 centims. long, and the part to the right of V, 81 centims. long, were made of strong brass tubing. The portion DD' between these was 20 centims. long, and consisted of an aluminium tube, which was of the same internal diameter as the brass cylinders. Brass collars over the ends of the aluminium tube fitted into the external collars V and V soldered to the brass cylinders, and so formed close-fitting joints that were made gas-tight by sealing them on the outside. The whole cylinder was supported on a board, XX', and insulated by means of four paraffin blocks, two of which are represented by P and P'. The inner cylinder, BB', was an aluminium tube 1 centim. in diameter, closed at its ends by conical pieces. At C the cylinder was divided so that the two portions were held one-half of a millimetre apart and insulated, by means of an ebonite plug. At the end, B', the tube was supported and kept central by means of two small ebonite rods, Q. The tube was further supported by the two stiff brass wires, Y and Y', which lead through the ebonite plugs, R and R', in the outer cylinder, and served to Digitized by Google 198 MR. J. ZELENY ON THE VELOCITY OF THE IONS make electrical connections. The part B' was joined to earth, while the part B w^as connected to a pair of the quadrants of the electrometer, E. Great care was taken to adjust the position of the central cylinder so as to be accurately concentric with the outer one. The ends of the outer cylinder were fitted with the large rubber stoppers F and F'. Through these passed the gas inlet and outlet tubes, whose ends were the elongated funnels J and J'. These funnels, together with the cone endings of the inner cylinder, made the lines of gas motion change less abruptly on entering and leaving the apparatus, and so aided in having the gas maintain a steady motion in DD', where the observations were taken. At the left end, F, a rubber tube led to a gas bag of about 150 litres capacity. The manometer, I, measured the pressure of the gas in the apparatus. The right end, F', was connected to the glass wool chamber, G, which served to remove dust and any stray electrification from the gas. A rubber tube then led to a drying or moistening apparatus, to be described later, which was connected to a large gasometer of the ordinary type. The pressure of the gas in the gasometer was measured by means of a manometer, and a scale was also attached to the gasometer for measuring its rate of descent during an experiment. The average velocity of the gas stream in the apparatus was determined from the volume emitted by the gasometer in a second, and from the area of the cross section between the two cylinders. To prevent the gas in the gasometer from getting moist too rapidly in those cases where dry gases were used, the surface of the water was covered with a layer of oil, such as is used for air pumps, because of its very low vapour pressure. The board, XX', with the attached cylinders was placed on the top of a lead- covered box, UU', so that DD', the aluminium portion of the outer tube, was above the aluminium window, W, in the box. The box contained the Crookes' tube and the induction coil for operating it. The form of tube used was that which the writer has previously employed for similar work.* This form was more satisfactory than any of the others tried, and gave the best results when emitting weak rays, and when an interval of rest of at least three or four minutes was allowed between the periods of use, which did not exceed thirty seconds. A 6-inch Apps' coil was used with a hammer interrupter, which could be made to run with sufficient uniformity with an easy running weak ray tube. The source of the rays, T, was more than 20 centims. from the axis of the cylinders. The narrow vertical beam of rays wliich was sent up through the cylinders was regulated by adjusting the position of the tube, T, and of the lead plate, S, with its narrow slit, and of the two lead rings, L and U, which fitted over the cylinder, DD'. This adjustment was first made by geometrical arrangement, and then tested and completed with the aid of a fluorescent screen placed over the apparatus. The lead strips, H and H', served to restrict tlie window, W, and the lead cover, Z, prevented any rays or ionized gas from reaching the outside air of the room. ♦ J. Zeleny, a^hil. Mag./ July, 1898, p. 126. Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. I99 The quadi'ant electrometer, E, used for making the measurements was a small bicellular one, the needle of which was suspended by a quartz fibre, and charged through the liquid below by means of a battery of 160 small storage cells. One pair of its quadrants was joined by a wire to the part BC of the inner cylinder. Both the electrometer and the connecting wire were surrounded by an earthed metal case. The key, K, permitted the insulated quadrants to be connected to earth at any time. The capacity of the two quadrants and the part of the inner cylinder connected to them, together with the connecting wire, was about 53 centims. The sensibility of the electrometer was about 500 divisions per volt, with the scale at a distance of 130 centims. The potential of the outer cylinder A A' was maintained at any desired value by means of the battery of storage cells, N ; the arrangement of the extra cell, O, and the divided megohm, M, permitting the addition of a fractional part of a cell's voltage. By opening a stop-cock on the gasometer the gas was made to pass from the gasometer, through the apparatus, into the gas bag on the other side, at a rate which was regulated by the weights on the gasometer. It could then be forced back into the gasometer and used again. A large volume of gas is required for carrying out an experiment, and the method is therefore limited to a small number of gases that can be obtained in such quantities, and that do not act upon the materials of the apparatus. § 4. CORRKOTIONS AND PRECAUTIONS OBSERVED IN THE EXPERIMENTS. 1. It is essential for these experiments that in its motion down that part of the tube where the observations are being taken, tlie different portions of the gas should move in paths parallel to the axis of the tube, i.e., that the motion be uniform, and not turbulent with vortices. This condition depends upon the velocity of the gas stream. O. Reynolds has shown* that for motion in a cylindrical tube a fluid when started in a turbulent state wilj tend to assume a uniform motion with the parts moving parallel to the axis when for the fluid the average velocity is less than a critical value. V = A^ BpD' where /x is the viscosity of the fluid relative to that of water at 0°, p is its density, D is the diameter of the cylinder, and B is a constant. The value of B obtained was about 280 when D and V were measured in metres. Applying this constant to the gases used, for a cylinder of the diameter of the ♦ O. Reynolds, * Phil. Trans.,' A, 1883. Digitized by Google 200 ME. J. ZELENY ON THE VELOCITY OF THE IONS outer one in the apparatus, we obtain for the value of the critical velocity for air. about 55 centims. per second, and for hydrogen about 390 centims. per second. It is evident that in the apparatus used where there are two concentric cylinders, the maximum velocity consistent with a uniform motion must be considerably larger than if the gas were flowing through the outer cylinder alone. Nevertheless the largest value of the velocity used in any experiment was 25 centims. per second for air and 44 centims. per second for hydrogen. As these values are well within the limits given above for a cylindrical tube whose radius is equal to that of the outer one here used, the conditions for a stable motion are fulfilled. The entrance of the gas through a funnel-shaped aperture and its subsequent passage for a con- siderable distance through a uniform section allowed the motion to come to a per- manent state before it reached the place where the observations were taken. An experiment which was tried showed that by blowing a stream of air down a large glass tube and with a velocity greater than that used in these experiments, the gas assumed a motion parallel to the axis after it had traversed but a short length of the tube, as was made visible by the presence in the air of irregularly distributed ammonium chloride particles. 2. The volume of the gas emitted per second by the gasometer varied a little for different elevations of the gasometer, but there was a considerable range where it was quite constant, and this range only was used in making experiments, the rate of descent being determined in addition during each observation. Guide wheels pre- vented the tilting of the gasometer during its descent, and the readings on the attached scale could therefore be relied upon. The pressure of the gas was determined by a manometer attached to the gasometer, and the pressure in the apparatus was similarly obtained. The volume of the gas emitted by the gasometer per second was then reduced to the pressure in the apparatus, and dividing by the flow area in the tubes, the required value of U in equation (9) was obtained. 3. In order to understand more clearly the manner in which the values of A and X of equation (9) were determined, let us consider the following case. In fig. 3, CO represents a longitudinal section of the outer cylinder. DB is the Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 201 inner cylinder having the insulated juncture at ky the part D of the cylinder being connected to an electrometer. The gas stream is supposed to flow from right to left in the figure, and hdmn is the beam of rays. DB being at zero potential, suppose that when the potential of CC is at a certain value the ions going towards DB move in paths parallel to the line ah in the upper half of the figure. An ion starting from any point to the left of ak would reach the part D and so influence the electrometer, but as all of the ions start from the beam of rays to the right of ak^ all of them reach B. If the potential of CC is diminished so that the inclination of the ionic paths becomes hk, ions from the outermost rim of hdmn will just begin to reach the part D. By a certain decrement in the potential of CC the paths of the ions can be made parallel to dk, so that ions will reach D from a volume whose section is represented by the triangle hdg^ the width of the beam of rays being hd. By a decrement in the potential of CC equal to the last one, the volume from which ions reach D is increased by a volume whose section is seen from the figure to be nearly a parallelo- gram of about twice the area of the triangle hdg. Another equal decrement in the potential increases the volume by almost the same amount as the last. As the potential is diminished further, the rate of increase of volume gradually diminishes. So if we represent the potentials used by abscissas and the volimies from which ions reach D by corresponding ordinates, we obtain a curve, fig. 4, R / '^n p/a Cs, / n / <q J f ' a / ^ / £ 1 / w / / / 1 7 / / u ■yf b C I i i z i f Fig. 4. whose inclination to the axis of abscissas, as the potentials are increased, at first gradually increases (RS of fig. 4), then assumes a constant value (ST) and finally diminishes (TU) as the curve ends in the axis of abscissas. The point U corre- voL. oxer.— A. 2 D Digitized by Google 202 MR. J. ZELENY ON THE VELOCITY OP THE IONS spends to the inclination of the paths of the ions represented by hh of fig. 3, T corresponds to dk, and W to ek. As the paths change from ek to dk^ the diminution in the number of ions reaching D is equal to about twice the number that are getting to D in the latter case (W^ = 3 Td). If, therefore, the rate of diminution remained unchanged until ions just ceased to reach D, the change in potential required for this would be just a half of the change fi'om ek to dk or from dk to hk. Thus in the curve it is seen that by prolonging WT it reaches the axis at c, half way between 6 and d. This corresponds to a potential which would be required for an ion starting from c (fig. 3) the middle point of the beam of rays 6d, in order to have it just reach the juncture k in the inner cylinder. It is evident that the points T and U are not very sharply defined on an experi- mental curve, and hence cannot be determined as accurately as the point c, and so in practice the potential A of formula (9) has always been determined in this latter way. Evidently the value of X which is to be used with this value of A has to be measured from the middle of the beam of the rays where they cross the inner cylinder to the middle of the juncture k^ as aU ions reaching the middle point are drawn to D. The width of this juncture was only '05 centim. The width of the beam of rays was used as small as possible, and in most cases was •2 centim., this being a smaU part of the total distance X 4. In considering the distribution of the ions between the two cylinders while the conduction is going on, it is seen from the lower part of fig. 3 that supposing the external tube to be positive, the negative ions starting from 5 will describe a path somewhat like sw, so that all of the negative ions will be confined to the space wmnts. Similarly the positive ions starting from m will describe the path mk^ and all of the positive ions will be confined to the space kmnts. In the space where these two overlap, i.e., omnts, both kinds of ions will be present and recom- bination will take place, the number of ions per cubic centim. diminishing, there- fore, as we go from 5m to o. The space ovnn will be occupied by negative ions alone, and oks by positive ions alone. wm will usually be shorter than ks^ because as a rule the negative ions travel the faster in the same electric field. 5. Of the ions starting from m towards k all will not follow the path mi, but some, due to the motions assigned to them by the kinetic theory of gases, will difiuse to either side so that the distribution, along the path, of the ions which started from m will lie between the two dotted lines mr and mp. This effect will produce a distortion in such a curve as that shown in fig. 4, and to bring all of the ions to the part B of the inner cylinder will require a greater force than would be necessary if there were no diffusion. The effect of this disturbance upon the value of the ionic velocity obtained in the manner described is to give a result that is too small because the potential A obtained is too large. Moreover the amount of the diffusion depends upon the time required for the ions to travel between the two Digitized by Google PRODUCED IN GASES BY ItftNTGEN RAYS. 203 cylinders so that if we obtain values of the ionic velocity, in the manner already described, these will be the larger and nearer to the true value the smaller the time that is required for the passage of the ions across. If this time were zero, then evidently all diffusion effects would disappear. 6. The free charges that exist in the gas, where the ions of one sign predominate, tend to spread on account of the mutual repulsion of the charged carriers. This produces an eflFect similar to that of the diffiision just described. It increases with the time required for the ions to pass between the cylinders, but is less the smaller the density of the free charges, i.e., the weaker the Rontgen rays used and the narrower the beam of the rays. 7. The presence of these free charges in the gas also has an influence upon the intensity of the electrostatic field between the two cylinders. To diminish this effect a sensitive electrometer was used in making the observations, as this allowed the employment of a weak radiation so that the charges in the gas were of a small density. While it is not possible to make an exact calculation of the magnitude of this effect because of the unsymmetrical distribution of the ions, an approximation to it can still be obtained. Knowing the capacity of the receiving system and the charge received in a given time, and knowing the approximate velocity of the ions in the electric field and the approximate space occupied by the firee charges, the density of these charges can be obtained roughly, and their effect upon the electrostatic field can be computed. Computations of this kind made from the observations used for final results showed that the largest value of this correction made a diminution in the electro- static field of less than 1 per cent. In some experiments where a large inner cylinder was used the intensity of the electric field employed was less, the ions moved slower, and the density of the fi^ee charges was therefore larger and in some instances the above correction was perhaps nearly 2 per cent. In all cases an increase in the strength of the field itself diminishes the percentage value of the correction, while the simultaneous diminution in the density of the free charges reduces it still further. 8. The motion of these free charges through the gas also produces a motion of the gas itself, as the writer has previously shown.* The amount of this is, however, very smaU compared to the velocity of the ions, so that it cannot have an appreciable disturbing efiect upon the restdts of these experiments. 9. In conduction produced by Rontgen rays there is a noticeable faU of potential at the electrodes which diminishes the electric intensity in the intermediate space. As determined by the writer, t for conduction in air between two plates 1*2 centims. apart, this amounted to about 2 per cent, of the total potential difference for the * J. Zeleny, ' Proc. Camb. Phil. Soc.,' vol. 10, Pt. L, p. 13. t J. Zbleny, ' Proc. Camb. Plul. Soc.,' vol. 10, Pt. I., p. 21. 2D2 Digitized by Google 204 MR. J. ;ZeleNY on the velocity op the ions strength of rays used. For the much weaker radiation and the greater distance here used the correction does not perhaps exceed I'per cent. For gases other than air the effect has not been determined, and has been assumed to be no greater than with air. 10. J. Perrin'*^ has shown that when the Rontgen rays impinge upon a metal surface the ionization in the gas near it is increased by an amount depending upon the nature of the metal and upon the state of its surface. M. G. SAGNAct and P. Langevin have shown since that this is due to a secondary radiation started at the metal surface by the Rontgen rays. It is possible that the ions so produced are of a different nature from those produced by the direct rays, but in the absence of any evidence to that effect the much more probable case is assumed that the two kinds are identical. The effect of the secondary rays, therefore, is to produce an uneven distribution ot the ions in the space exposed to the direct rays, and also to widen the ionized area near the metal surfaces. This makes more difficult the accurate determination of the potential A in equation (9), the tendency being to get it too large. J. Perrin found that the surface effect was by far the least for aluminium, what he calls the coefficient being '0 for aluminium in air as compared to '9 for gold in air. The effect is also very much dependent upon the cleanliness of the surface. It is thus seen that in the apparatus used this effect was made as small as possible by using unpolished aluminium as the material for those parts of the cylinders upon which the rays impinged. That the secondary rays did not produce an appreciable amount of ionization at a short distance to the side of the beam of the direct rays was shown by passing these rays near to the insulated juncture in Ihe inner cylinder while the gas in the tubes was at rest. No conductivity was observed to that part of the inner cylinder which was not exposed to the direct rays. Further experiments tried for the effect of the secondary rays by coating the inside of the aluminium cylinder on the apparatus with tin-foil will be described later among the observations for dry air. 11. W. C. RontgenJ has shown that the air itself where it is exposed to the rays acts as a source of a weak secondary radiation. The writer is not aware of any experiments showing any conductivity produced by this radiation, but the experiment referred to in the last section, where a beam of rays near the juncture of the inner cylinder produced no appreciable conductivity on the other side, shows that in these experiments the effect may be disregarded. 12. When D (fig. 3), the part of the inner cylinder joined to the electrometer, takes up a charge in the progress of an observation, the electric field in the vicinity of the juncture becomes slightly distorted, tending to lessen the number of ions * J. Perrin, 'Comptes Eendus/ vol. 124, p. 455. t M. G. Sagnac, 'Journal de Physique,' 1899, p. 65. X W. C. EONTGEN, * Wied. Ann.,' vol. 64, p. 18. Digitized by Google PEODUCED IN GASES BY RONTGEN RAYS. 205 reaching D. As for each reading this effect starts from zero, the only influence of this upon a series of readings with difierent potentials is to diminish their values by small amounts nearly proportional to their size, thus having practically no effect upon the result obtained by projecting the curve as in fig. 4. 13. The velocity of the ions is evidently dependent upon the pressure of the gas. In these experiments the variations in the pressure were but small, being due mainly to the variations of the barometer. No experiments have been carried out on the effect of pressure upon the velocity of the ions produced by Rontgen rays, but E. Rutherford* has shown that for the conduction produced by ultra-violet light the velocities of the ions in air are inversely as the pressure of the gas. This result will be used in these experiments to reduce all of the values of the velocities to the same pressure of 76 centims. of mercury. 14. The effect of temperature upon the ionic velocity is not known, so that correc- tions for temperature could not be made. The temperature was, however, taken in all cases, so that if necessary the correction can be applied later on. 15. In considering the various corrections above, it is seen that the effect of many of them is diminished or made negligible by using a narrow beam of weak rays, and by using unpolished aluminium for that part of the cylinders where the rays impinge. Those corrections which depend upon the time required for the ions to cross between the two cylinders could be made very small by sufficiently reducing the value of this time, but we are limited in doing so by the increase that is produced in the difficulty of measuring one of tlie required quantities. Resort must be had to finding the values of the ionic velocities for different times of crossing, and from these deriving the final results. An estimated correction of 2 per cent, will be made for those effects considered above, especially (7) and (9), which tend to make the result too small by an undeter- mined but small amount. § 5. Changes made in Experimental Conditions. The apparatus used permits of several changes in the experimental conditions, which are a test of the accuracy of the method, and allow us to draw conclusions about the effects of some of the corrections previously noted. 1. The velocity of the gas stream was varied by changing the weights on the gasometer. This necessitated a proportionate change in the value of the potential A of equation (9). The paths described by the ions are the same, but the time required for their passage between the two cylinders is changed. There are also changes in the amount of recombination of the ions and in the diffusion effect. The density of the free charges is changed, and so their effect upon the electric intensity is altered, and the spreading due to the mutual repulsion of the ions is also different. * E. KirniERFORD, * Proc. Camb. Phil. Soc.,* vol. 9, Pt. VIII., p. 4U. Digitized by Google 206 . MR. J. ZELENY ON THE VELOCITY OP THE IONS 2. The distance of the beam of rays from the insulated juncture in the inner cylinder was also changed. This likewise necessitated a change in the value of the potential A, but in the opposite sense. The paths of the ions are now quite different, and changes are also produced in all of the quantities mentioned in the preceding case. 3. The intensity of the Rontgen rays was also varied. This produced alterations in the density of the free charges in the gas, and consequently in their effect upon the electric field between the cylinders and in the mutual repulsion of the ions. The amount of the recombination of the ions is also affected as well as the fall of potential at the electrodes. 4. By changing the diameter of the internal cylinder complete changes are produced in the configuration of the forces, and of the motions of the ions. All the other changes can also be tried in conjunction with this one. 5. The material of the inner surface of the outer cylinder was also altered to note the influence upon the result of increased ionization at the metal surface. 6. In trying to find the effect of any of these changes upon the observed velocity the greatest difficulty met with is due to the smallness of the effects, and their conse- quent masking by the irregularities of individual observations caused by the difficulty of maintaining a uniform radiation for a length of time sufficient to cover a niunber of readings. Individual observations taken under the same conditions may vary among themselves by a number of per cent., so a smaU change in the result cannot be detected unless a large number of observations is made. § 6. Method of Conducting the Experiments. The following procedure was followed in taking readings with the apparatus. The Crookes' tube and the lead slits were accurately adjusted, so that the beam of rays occupied the desired position, and the distance X of equation (9) was carefully measured. The cylinder AA' was connected to a chosen potential on the battery N. The electrometer quadrants, joined to the part B of the inner cylinder, were then disconnected from earth by means of the key K, and the zero reading was observed on the scale. The reading on the gasometer scale was also taken. At a definite time, observed on a chronometer, the valve at the gasometer was opened, so that the gas began to flow through the apparatus. After a short period, usually 10 seconds, sufficient to produce a steady state of flow in the apparatus, the primary of the induction coil was closed and the rays thus started. The rays were allowed to run for 30 seconds, and the primary of the coil was then broken, and the valve of the gasometer was also closed at a definite time. The electrometer reading was now taken, and the deflection produced was obtained. The key K was then closed, and the quadrants of the electrometer were connected to earth. From the reading on Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 207 the gasometer scale the volume emitted was obtained, and with the aid of the pressure readings which were taken the average velocity of the gas stream in the apparatus could be calculated. An interval of about three minutes was allowed as a rest for the tube, as this made it much more constant over a large number of readings. Iii the mean time, if necessary, gas was forced back from the gas bag into the gasometer. Guided by the previous electrometer deflection the potential of the outer cylinder was now changed, and the whole process repeated. In this way a number of readings were taken, such that the electrometer deflections ranged from some value down to near zero. These were taken in such an order that at first, say, a descending series of readings was obtained, and then immediately afterwards an ascending series. In this manner it is possible to detect any uniform changes which are taking place in the intensity of the rays, for in that case the two series of points would lie on curves of different inclinations. It was seen in § 5 that the time of passage of the ions from one cylinder to the other could be varied by changing the velocity of the gas stream, and also by changing the distance X. Both of these were employed in practice, and it was found that the values of the velocity obtained diminished as the time increased ; but they were practically the same for two different values of X if the velocity of the gas stream was changed in the same ratio, i.e., if the time of passage of the ions was the same. J, S. TowNSEND* has recently observed that the rate of diffusion of the ions depends upon the moisture in the gas. In these experiments the gases were used both dry and saturated with aqueous vapour, and it was found that the velocity was different in the two cases. For saturating a gas with aqueous vapour it was forced, in passing between the gasometer and the apparatus, to bubble through a water bottle and then to pass through a long horizontal tube half filled with water. After the gas had been passed several times back and forth between the gasometer and the gas bag, and before any readings were taken, the water bottle was cut out so as to avoid any unsteadiness in the pressure due to the bubbling. For drying a gas the above arrangement was replaced by one in which the gas had to pass through a long, horizontal glass tube, partly filled with concentrated sulphuric acid, and then through a large volimie of calcium chloride. In order to allow a suflBciently rapid stream with the small pressures used the calcium chloride was placed in a large, wide bottle, the gas entering above and leaving by a protected ftmnel-shaped tube near the bottom. It thus had to traverse a considerable length of calcium chloride, and on account of the large area of the bottle the velocity through it was smaU, ♦ J, S. TowNSBND, 'Phil, Trans.,' A, vol. 193. Digitized by Google 208 MR. J. ZELENY ON THE VELOCITY OF THE IONS § 7. Moist Air. The following is an example of a set of readings taken for the positive ions in air saturated with aqueous vapour. Letters refer to corresponding quantities in formula (9). Temperature = 14*5° C X = 2'60 centims. a = '50 centims. b = 2 '555 centims. Width of beam of rays = '20 centim. Barometer =75*4 centims. Excess pressure inside gasometer =1*56 centims. of mercury. „ „ in apparatus = '59 centim. of mercury. 20 cells = 42 "6 volts. Table I. — Moist Air. Positive Ions. Voltage of outer cylinder. • Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. Cells. Divisions. Centims. + 10 145 6-77 + 12 105-5 6-79 + 14 68-5 6-78 + 16 29-5 6-72 + 18 + 19 12 6-70 6-83 7 + 17 19 6-81 + 15 52-5 6-78 + 13 87 6-77 + 11 128 6-76 In the middle of the observations the gasometer was refilled from the gas bag. The sectional area of the gasometer was 2904 sq. centims., and the area between the two cylinders was 19*73 sq. centims., so the average rate of descent of the gasometer above indicates an average velocity in the apparatus of 25*2 centims. per second, when corrected for the difference in pressure between the gasometer and the apparatus. The voltages and their corresponding deflections are represented graphically in curve I. of fig, 5. The set of readings here given, and most of those which are to follow as examples, have been selected from among the best obtained. It is seen that the curve at first approaches the axis of abscissas in nearly a straight line, but becomes convex when near to it. Had readings been taken for voltages smaller than those used, that part of the curve would have been concave to the axis of abscissas. It has been explained in § 4 (3), why there is a nearly straight portion in the curve, while the width of the beam of rays and the various causes tending to spread Digitized by VnOOQ iC PRODUCED IN GASES BY RONTGEN RAYS. 209 the ions make the lower end of the curve approach the axis at a less rapid rate. It was also shown that the point on the axis of abscissas, obtained by prolonging the straight portion of the curve, would correspond to the voltage required to make an / / / } J. - /40 / / r / J -i£0 x^ / f n 1 / irZJ__ — 900 S / ^ - 'S / 7 _ -ao^ / / -^ 1 J' — AQ ^ / 1 / r ~ 40 S i / / ,-2 i 2 1 / — 20 fr- /' -J F^ ^ jr- ^ 1 .0 ?0 1 1 6 VoU /4 ! in 1 'ig. 5. a ^ ion, starting from the surface of the outer cylinder in the middle of the beam of rays, just reach the middle of the juncture in the inner cylinder. But with diffusion and the other causes acting to produce a spreading of the ions, it is evident that the inclination of the straight part itself is affected and the result changed. Corrections for this error can only be made in conjunction with those of some other effects, and that by experiment, by producing alterations in the amount of these effects, by changes in the time of passage of the ions across the space between the cylinders. The velocity obtained by the use of the voltage determined by the continuation of the straight part of the curve, as shown in the figure, will be called the ionic velocity for that determination, it being tmderstood that it is not implied thereby that the velocity changes with the time, but that this is only a step towards the final result. From the above curve, A is seen to be 177 cells, which is equal to 377 volts. Using equation (9), V = j — 5 — log<f~ f Tv = 5'118 ^^ — :^^ = 1'315 centims. per second, The pressure in the apparatus is 76 centims. of mercury VOL. cxcv.— A, 2 E Digitized by Google 210 MR. J. ZELENY ON THE VELOCITY OF THE IONS From equation (10) '*^ = i=ll = 'i*^^^"**- The following is a set of readings taken for negative ions in moist air. Unless otherwise mentioned the values of a, 6, and the width of the beam of rays will hereafter be taken the same as in the previous example. Temperature = 14*4° C. X = 6*42 eentims. Barometer = 74*7 eentims. Excess pressure inside gasometer = 1'54 eentims. of mercury. „ „ in apparatus = '59 centim. of mercury. 8 cells =16-5 volts. Table II. — Moist Air. Negative Ions. Voltage of outer CTlinder. Electrometer deflection Descent of gasometer in 30 seconda. in 40 seconds. Cells. Division*. Centims. -4 128 605 -5 • 68-5 5-95 -5-4 45 5-94 -6 17-5 5-92 -5-6 32-5 5-89 -5-2 50 5-90 -6 18 602 -5-4 44-5 5-99 -5 67-5 5-96 -4-4 95 5-90 -7- 2 5-90 The results are represented in Curve II. of fig. 5. U = 22*1 centims. per second. A = 127 volts. 221 = 1'39 centims. per second. The pressure in the apparatus =75*3 centims. The velocity reduced to 76 centims. pressure = 1'38 centims. per second. T = ^ = -29 second. The following is a summary of the results obtained for moist air for both the positive and negative ions. Each result was obtained from a series of observa- tions as indicated by the above examples. The results are reduced to 76 centims. pressure. Letters refer to quantities in equations (9) and (10). Digitized by VnOOQ iC PRODUCED IN GASES BY RONTGEN RAYS. 211 Tablk TTT. — Moiflt Air. Summary of Besult£ ). Ionic velocity. Reference number. X. U. A. T. Tempera- ture. Gas pressure. Negative. Positive. 1 4-33 20-6 + 19-2 -21 •c. 15-3 76-2 1-28 2 4-18 10-85 + 11-1 -39 15 77-6 — 1-225 3 4-18 111 -10-1 •38 15 77-6 1-37 — 4 418 10-73 - 9-8 •39 14-3 76-5 1-35 — 6 4-18 10-96 + 11-35 -38 14-3 76-5 — 1-19 6 4-18 25-0 + 23-4 •17 14-3 76-8 — 1-32 7 418 25-0 -21-2 -17 14-3 76-8 1-46 — 8 2-68 11-3 - 15-65 -24 14-6 75-9 1-38 — 9 2-68 11-3 + 17-5 -24 14-6 75-9 — 1-23 10 2-68 22-0 + 32-15 -12 14 76-1 — 1-32 11 2-68 221 -29-1 -12 14 76-1 1-46 — 12 8-41 11-33 + 6-26 -75 14-4 76-0 — 110 13 8-41 11-23 + 6-03 -75 14-4 76-0 — 113 14 8-41 10-78 - 5-2 -78 14-4 76-0 1-26 — 15 8-41 11-33 - 5-41 -74 14-4 76-0 1-27 — 16 8-41 11-8 - 5-51 -72 14-4 760 1-30 — 17 8-41 24-8 - 10-95 -34 14-2 76-2 1-39 — 18 ' 8-41 24-73 + 12-2 -34 14-2 76-2 — 1-24 19 8-41 24-8 + 12-5 -34 14-2 76-2 — 1-22 20 8-41 24-8 -10-7 -34 14-2 76-2 1-42 — 21 6-42 10-67 - 6-57 -60 13-5 75-8 1-30 — 22 6-42 10-7 + 7-26 -60 13-5 75-8 — 1-175 23 6-42 22-06 + 14-0 -29 14-4 74-8 — 1-245 24 6-42 22-06 + 14-15 •29 14-4 74-8 — 1-23 25 6-42 22-1 -12-7 -29 14-4 74-7 1-38 — 26 2-60 11-1 -15-7 -29 14-7 74-7 1-375 — 27 2-60 111 + 17-0 -24 14-7 74-7 — 1-27 28 2-60 25-2 + 37-7 •10 14-5 75-4 — 1-315 . 29 2-60 25-2 -34-1 •10 14-6 75-4 1-466 — 30 2-60 12-6 -17-8 -21 15 75-5 1-39 — 31 2-60 10-9 -15-7 -24 15 75-5 1-37 — 32 2-60 13-1 -18-3 •20 16 75-7 1-41 — 33 2-60 1316 -17-8 •20 16 75-7 1-45 -^ .7 .a ^ .3 T. In aeconds. Fig. 6. 2 E 2 .£ Digitized by Google 212 MR. J. ZELENY ON THE VELOCITY OF THE IONS The results are represented iu fig. 6, where the velocities are represented as ordinates and the corresponding values of T as abscissas. It is seen that the velocities decrease with T, and nearly in a linear manner. Considerable variations are observed among the individual results, but it is believed that they are not greater than is to be expected from the nature and difficulties of the experiments. In § 4 it was seen that some of the corrections which act to give too small a value for the velocity diminish with T and disappear for T = 0. By drawing lines through the points in fig. 6 and prolonging them to the axis of ordinates where T = 0, we obtain the most probable values of the velocities. This gives for the negative ions 1*48 centims. per second, and for the positive ions 1*34 centims. per second. In § 4 (15) it was stated that a correction of 2 per cent, would be made for dis- turbances not corrected by the above method. This gives for the final results for moist air the velocity in an electric field of 1 volt per centim. for the negative ions =1*51 centims. per second, and for the positive ions = 1'37 centims. per second at a temperature of about 14° C, and a pressure of 76 centims. of mercury. §8. Dry Air. The following set of readings was taken for the positive ions in dry air : — Temperature = 13'8° C. X = 2'60 centims. Barometer = 761 centims. Excess pressure in gasometer = 1 "6 centims, of mercury. „ „ apparatus = '45 „ „ 14 cells = 29-0 volts. Table IV. — Dry Air. Positive Ions. Voltage of outer cylinder. Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. Cells. Divi«ioiia Centims. 1 + 8 117 4-29 + 10 60 4-28 ! + 12 22 4-26 + 14 7 4-26 + 11 40 4-22 + 9 94 4-2.5 + 7 153 4-25 + 10 62 4-21 ! + 8 123 4-23 These results are represented graphically in Curve I. of fig. 7. The corrected value of U is 15 '9 centims. per second. A = 24-8 volts. 15-9 Sov = 5-118 2-60 X 24'8 = 1*26 centims. per second, and when reduced to 76 centims. pressure this becomes 1'27 centims. per second. rv 2-60 ,^ J T = — = -16 second. Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. The following set of readings was taken for the negative ions in dry air : — Temperature = 15'8° C. X = 2*60 centims. Barometer = 76 centims. Excess pressure in gasometer = TO centim. „ „ „ apparatus = '13 centim. 7 cells =14-5 volts. 213 Fig. 7. Table V. — Diy Air. Negative Ions. Voltage of outer cylinder. Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. Celli. DiTisions. Centims. -2 141-5 2-07 -3 87 2-05 -4 28-5 208 -6 6-5 2-05 -4-4 18 2-08 -3-4 63 205 -2-4 116 205 The results are shown graphically in Curve IL of fig. 7. U corrected for pressure = 7 '64 centims. per second. A = 9-21 volts. Digitized by VnOOQ iC 214 V = 5-118 MR J. ZELENY ON THE VELOCITY OF THE IONS 7-64 2-60 X 9-21 — = 1*63 centima per second. T = - If? = -34 second. A summary of the results obtained for dry air for both the positive and the negative ions is given in Table VI. Table VI. — Dry Air. Summary of Results. Ionic velocity. Beference X. U. A. T Tempera- Gas number. J- • ture. pressure. Negative. Positive. I 2-60 7-64 - 9-21 34 °c. 15-8 76-2 1-63 2 2-60 7-64 + 120 34 15-8 76-2 1-25 3 2-60 7-64 + 12-2 34 16-3 75-8 1-23 4 2-60 7-57 - 9-31 34 16-3 75-8 1-60 5 2-60 7-28 - 915 36 12-2 76-6 1-58 6 2-60 7-23 + 12-15 36 12-2 76-6 M8 7 2-60 15-8 - 18-3 16 13-8 76-6 1-71 8 2-60 15-9 + 24-8 16 13-8 76-6 — 1-27 9 2-60 15-9 -18.1 16 13-8 7«-6 1-74 10 2-60 15-9 -18-3 16 .13-8 76-6 1-725 11 2-60 16-6 - 18-65 16 14-6 75-8 1-75 12 2-60 16-3 - 18-22 16 14 76-8 1-78 13 2-60 15-5 + 24-8 17 12-5 76-7 1-25 14 2-60 15-5 + 23-7 17 10-7 77-6 1-31 15 2-60 15-6 -18-1 17 10-7 77-6 1-72 16 515 8-65 + 7-76 60 11-4 77-3 — 1-13 17 515 8-62 - 5-9 60 11-4 77-3 1-47 18 5-15 15-6 - 9-31 33 11-4 77-6 1-67 19 515 15-7 + 12-6 33 11-4 77-6 1-26 20 515 8-58 + 7-64 •60 11-7 77-5 _ 114 21 5-15 8^58 - 5-88 60 11-7 77-5 1-48 At No. 7 the drying apparatus was changed, and at No. 12 the Crookes tube was replaced by a new one. The results are represented in fig. 8, excluding the points marked by squares. The final values thus obtained for dry air when the 2 per cent, correction men- tioned in § 4 (15) has been added, give the velocity of the negative ions = 1 '87 centims. per second, and of the positive ions = 1*36 centims. per second. The temperature varied several degrees between the difierent observations, but was on the average about 13° C. Most of the tests to which the method used in these experiments was subjected by changes of experimental conditions, were tried with dry air. Among these was tried the effect of changes in the intensity of the rays. By interposing aluminium plates the rays were diminished so that the conductivities produced by them changed Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 215 in the ratio of three to one, but no noticeable change in the result could be observed. During the course of all of the experiments the rays were not of the same intensity, for the Crookes' tube had to be replaced sevei'al times, but in aU cases without any marked effect upon the values obtained. It must be said, however, that rays of great intensity were never employed, the aim being always to have them as weak as possible for reasons previously stated. '^^~ o nl 1 ^ LB t ^ % O a /ii ^ Vk ?. S^ ^ /.6^ o _^^ , ^ J t 1 •^ X — M^ -TT ^ fib Stt ve Lit- ^ /Vi Ur- — X '^1 r- .t > .i ■> A % 7 I -is- eci 8. )/7fl to. \i % — ' > J f — ' The most severe teat to which the method was subjected was a change in the dimensions of the inner cylinder. In the above experiments the diameter of this cylinder was 1 centim., and it was now exchanged for one having a diameter of 2 '8 centims. The distance between the inner and the outer cylinders was thus diminished to nearly one-half of its former value. The electric field between the two became much more uniform, and the gas velocities for different points of a cross- section now varied in a different manner. In order to keep the other quantities the same, the small distance between the two cylinders necessitated the use of voltages only about one-quarter as large as those used in the former arrangement. This increased the difficulty of the measurements and also some of the corrections which must be applied to get the final result. The density of the free charges in the gas was greater because the ions moved slower, being in a weaker field, and the same fall of potential at the electrodes was a larger percentage of the total voltage. The width of the beam of rays used was '3 centim. The following is a summary of the results obtained : — Digitized by Google 216 MR. J. ZELENY ON THE VELOCITY OF THE IONS Table VII.— Dry ■ Air. Summary for Large Inner Cylinder. Ionic velocity. Beference number. X. U. A. T, Tempera- ture. Gas pressure. Negative. Positive. 1 5-4 10-7 -1-90 -50 "C. irs 75^9 143 _ 2 5-4 12-6 -2-23 •43 124 759 143 — 3 5-4 12-7 + 2-84 •43 124 759 — 113 4 5-4 10-4 + 2-35 -52 12^4 75^9 — 110 5 3-77 11-55 + 3-75 -33 15 75^8 112 6 3-77 11-55 -2-70 •33 15 75^8 1^555 7 3-77 11-65 -2-80 -33 15 75^8 151 8 511 13-9 -2-64 -37 154 76^8 1^43 9 5-11 13-95 + 3-39 •37 15^4 76^8 1115 10 3-26 13-8 -406 •24 15^4 76^8 157 11 6-36 8-83 -1-48 •72 16 77^2 1315 12 6-35 8-83 + 1-80 -72 16 77^2 — 1-08 13 6-36 14-05 + 2-80 •45 15^8 76^0 — 1-09 U 6-35 14-2 -2-20 •45 15^8 760 140 15 2-63 14-9 -4-7 •18 15^8 76-0 165 16 2-63 7-7 + 3-5 •34 163 75^9 — M4 17 2-63 7-04 r2-5 •37 163 75^9 1-465 18 2-63 13-9 -4-32 •19 16-3 759 167 19 2-63 14-06 + 5-94 •19 16-3 75^9 1-23 The results are represented in fig. 9. y^ ^ ^ l^ y^ J ^^^ aC Ve v^ ^ O ^ ^ o ^ ^ ^ o ( ) ^ ^ ^ o »**■ «,- -- — •*' — • »•* - ^ i ^ m fe. J^ ^ ^ ^^ ^ » X -" M— — ►— ■■"" "•' k-- i-H r^ •7 f-r € f s \ . - ^ f •<s \ I 1 . ./ \ fl r in seconds. Fig. 9, hS 1.7 i.5 $ .^ 1.3 :2 J ht Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 217 It is seen that in this case the values of the velocity change less rapidly as the values of T become large ; but for the smaller values of T the change is more rapid than it was when the smaller inner cylinder was used. The points on the curves are not advantageously distributed, and so do not allow of a very accurate projection of the lines to T = ; but from those drawn it is seen that the results are but slightly smaller than those obtained with the smaller inner cylinder. This is considered a good agreement even if it is left out of account that an additive correction is still to be made. An alteration which was tried to test the eflfect of surface ionization was a change in the material of the inner surface of the outer cylinder. The aluminium part DD' (fig. 2) of the outer cylinder was coated on its inner surface with a layer of tin-foil. The rays in penetrating the cylinder now had a tin instead of an aluminium surface in contact with the air. J. Perrin has shown (see § 4 (10) ) that what he calls the coefficient of the increased ionization at a metal surface is for tin in air '6 as against •0 for almninium in air. The effect varies with the state of the surface. In these experiments the aluminium surface used was an ordinary unpolished surface, while the tin surface used was that of bright tin-foil. It was thought that if an increase of the ionization near the metal surface has any marked effect upon the value of the velocity obtained, the difference should be observed by this new arrangement. The results obtained are given in Table VIII. , dry air being used as before. The smaller inner cylinder having a diameter of 1 centim. was used. Table VIII. — Dry Air. Summary for Tin Surfiice. Ionic velocity. Beference number. X. U. A. T. Tempera- ture. Q&8 pressure. Negative. Positive. 1 6-22 8-56 - 606 •6 "C. 139 77-3 142 2 6-22 8-61 + 7-82 •6 139 773 — 1105 3 6-22 18-2 + 15-2 •29 14-4 77^4 — 1-20 4 5-22 18-2 -10-8 •29 14-4 77^4 1^68 5 2-62 17-3 -19-6 •15 » 77-2 176 6 2-62 18-5 -21-2 •14 16-5 77^2 173 — 7 2-62 18-5 + 29-4 •14 16^5 77^2 1-25 The points are plotted as squares on the curves in fig. 8, which represent corre- sponding values when the aluminium surface was used. It is seen that the points for the negative ions agree very well with the curve. The points for the positive ions are 2 to 3 per cent, below the values for the aluminium surface. Taking both results into consideration it was concluded, if the addition of a tin surface changed the values of the velocities by but such a small amount, that originally when the VOL. OXCV. — ^A. 2 F Digitized by Google 218 MR. J. ZELENY ON THE VELOCITY OF THE IONS aluminium surface was used, the eflfect of the surface ionization could not have been sufficient to produce any marked error in the results. The surface ionization also varies with the nature of the gas, but the values obtained by J. Permn for aluminium with the gases used in these experiments were in all cases much less than for tin in air. § 9. Oxygen. The gas used in these experiments was the commercial oxygen obtained from a cylinder, which contained about 5 per cent, of impurities, mostly nitrogen. Since the size and nature of the apparatus prevented the employment of the most pure gases, it seemed advisable to use the cylinder gas. The density was changed but little by the presence of the impurities, and, so far as known, the velocity should therefore be but slightly aflfected. The drying of the gas and its saturation with aqueous vapour were carried out in the same manner as with air. The following set of readings was taken for the negative ions in oxygen saturated with aqueous vapour : — Temperature = 17-3° C. X : Excess pressure in gasometer „ „ apparatus 8 cells = 16-3 volts. 5*01 centims. 1*54 centims. •55 centim. Barometer = 76*4 centims. Table IX. — Moist Oxygen. Negative Ions. Voltage of outer Electrometer deflection Descent of gasometer cylinder. in 30 seconds. DivisionB. in 40 seconds. Cells. Centims. -3 194-5 6-06 -4 157 511 -5 106-5 5-02 -6 41 511 -6-6 17 5-07 -5-6 67 5-08 -5-2 93-5 509 -4-6 125-5 5-03 The results are shown in Curve I. of fig. 10. The corrected value of U = 1883 centims. per second A = 13-55 volts. 18-83 V = 5-118 = 1*413 centims. per second, and when reduced to T = 501 X 13-55 76 centims. pressure this becomes 1'43 centims. per second, 5-01 18-83 = '27 second, Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. J 7^ OJU / / / 1 / ( t f60 / I \ 1 i f J / i t / J /«)^ ' f f ^ J ■»s 1 J M. A^ C^ 1 4 oO s 1 a ) r . J J A 1 40 ^ y n / /^ 1 1 1 ; f t Vo Ltc t in a F 10. i > 219 The following is a set of readings taken for the positive ions in oxygen saturated with aqueous vapour : — Temperature = 15*6° C. X = 5'01 centims. Barometer = 76-5 centims. Excess pressure in gasometer = '44 centims. „ „ apparatus = '16 „ 6 ceUs = 12-35 volts. Table X. — Moist Oxygen. Positive Ions. Voltage of oater Electrometer deflection Descent of gasometer cylinder. in 30 seconds. in 40 seconds. Cells. Diviaions. CentiniB. + 2 US 2-30 + S 85-6 2-31 + 3-6 20-6 2-29 + 3-2 41 2-29 + 2-8 73-6 2-27 + 2-4 112 2-25 + 2-2 127 2-26 The results are represented by Curve 11. of fig. 10. 2 F 2 Digitized by Google 220 MR. J. ZELENY ON THE VELOCITY OF THE IONS The corrected value of U = 8 "42 centims. per second. A = 7-42 volts. V = 5-118 8-42 = 1'16 centims. per second, and when reduced to 5-01 X 7-42 76 centims. pressure this becomes 1'17 centims. per second. T = —r^ = '6 second. o'4^ A summary of the results thus obtained for moist oxygen for both the positive and the negative ions is given in Table XI. Table XI. — Moist Oxygen. Summary of Results. Beference number. X. U. A. T. Tempera- ture. Gas pressure. Ionic velocity. Negative. Positive. 1 2 3 4 5 6 7 8 9 10 6-89 6-89 6-89 6-89 5-01 5-01 501 5-01 3-03 3-03 9-41 9-38 16-8 16-8 8-42 8-68 18-8 18-6 20-5 20-6 + 604 - 5-38 - 8-80 + 10-5 + 7-42 - 6-69 - 13-55 + 15-7 + 27-3 - 24-25 •73 •73 •41 •41 •6 •58 •27 •27 •15 •15 "C. 15^4 154 156 15-5 15-6 15-6 17-3 17-3 15-4 15-4 76-8 76^8 773 773 76^7 76^7 769 769 76-9 76-9 1-31 1-44 1-34 1-43 1^46 117 fil 117 1-23 1-29 The results are shown graphically in fig. 11. .6 .5 T in seconds. Fig. 11. .3 Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 221 The correction mentioned in § 4 (15), which is to be applied to the values indicated at T = has in this case been reduced to 1 per cent., 1 per cent, being allowed for an increase in the velocity due to a diminution of density caused by the impurities in the gas. The corrected value thus obtained for the velocity in moist oxygen is for the negative ions = 1'52 centims. per second, and for the positive ions = 1*29 centims. per second, at a pressure of 76 centims. and at a temperature of about 16° C. The following is a summary of the results obtained for the positive and negative ions in dry oxygen : — Table XII. — Dry Oxygen. Summary of Results. Beference number. X. U. A. T. Tempera- ture. Gas pressure. Ionic velocity. Negative. Positive. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 2-73 2-73 2-73 2-73 2-73 3-89 3-89 3-89 3-89 3-89 6-89 6-89 6-89 6-89 13-9 13-8 13-3 13-3 13-3 16-65 17-7 16-9 7-97 8-46 8-92 9-07 17-35 17-4 + 20-2 - 15-37 + 19-5 -14-7 + 19-5 -13-6 + 17-7 -13-4 - 6-82 + 9-45 + 5-64 - 4-63 - 8-06 + 10-33 -20 -20 -21 •21 -21 -23 -22 -23 •49 •46 -77 -76 -40 -40 "0. 20-3 20-3 19-4 19-4 19-4 16 15-6 16-6 16 16 15-2 15-2 15-8 15-8 77-3 77-3 77-4 77-4 77-4 78 78 78 76-8 76^8 77-1 77-1 77-1 77-1 1-71 1-72 1-66 1-71 1-56 fsi 1-63 1-31 1-31 1-30 1-35 1-19 1-19 1-27 The results are represented graphically in fig. 12. When, as in the case of moist oxygen, a 1 per cent, additive correction is applied to the values indicated in the figure by T = 0, the final result for the velocity in dry oxygen is for the negative ions = 1'80 centims. per second, and for the positive ions = 1'36 centims. per second for a pressure of 76 centims. and a temperature of about 17^ C. Digitized by Google 222 MR. J. ZELENY ON THE VELOCITY OF THE IONS — 1 ' — — /.y Ni ^ Civs. n Ol^ ^^ ■^ /•7 ^ . -^ ^ >— i— ^ — /:3 Pd Sit ive 4^ . ■ — J ^ — — /%3 _ ^ — ^ /•/ • t f A 5 .1 J ,* 4 .J .£ J r I * I in seconds. Fig. 12. § 10. Carbonic Acid. The gas used was taken from a cylinder of liquid carbonic acid. The small amount of impurities in this does not produce any marked change in the density of the gas, and is assumed to be without noticeable effect upon the ionic velocities. As examples of the readings taken, the following two sets are given for carbonic acid gas saturated with aqueous vapour : — Temperature = 16'3° C. X = 3-02 centims. Excess pressure in gasometer = '44 centim. „ „ „ apparatus = '21 „ 10 cells = 20-6 volts. Barometer = 75*4 centims. Table XIII, — Moist Carbonic Acid. Negative Ions. Voltage of outer cylinder. Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. Celb. DiTisions. Centims. - 6 136 2-34 - 7 106-5 2-33 - 8 76 2-33 - 9 41-5 2.30 -10 17-5 2-33 - 9-4 28 2-29 - 8-4 57 2-28 - 7-4 91-5 2-27 Digitized by VjOOQ IC PRODUCED IN GASES BY RONTGEN RAYS. The results are shown in Curve I. of fig. 13. 2'23 lao VolCcL^e in ceils. Fig. 13. The corrected value of U = 8*52 centims. per second. A = 21'1 volts. '302 n4 ~ ^^^ centim. per second, which, reduced to 76 V = 5-118: centims. pressure, becomes '679 centim. per second. m 302 „^ J T = T7^ = -36 second. Table XIV. — Moist Carbonic Acid. Positive Ions. Voltage of outer cylinder. Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. OelU. Divisions. Centims. + 2 176-5 2-32 + 3 170 2-31 + 6 106 2-33 + 7 77-5 2-32 1 +8 45 2-32 + 9 17 2-32 + 8-4 28 2-26 + 7-4 62 2-25 + 6-4 i 97-5 2-27 Digitized by VjOOQ IC 224 MR. J. ZELEITS ON THE VELOCITY OF THE IONS These results are represented in curve II. of fig. 13. The corrected value of U = 8 "48 centims. per second. A = 19-16 volts. 8-48 .= "749 centim. per second, which reduced to 76 centims. " - ^'^^^ 302 X 1916 pressure becomes "745 centim. per second. 3*02 T = r-^ = -36 second. o.4o A summary of the results for moist carbonic acid is given in Table XV. Table XV. — Moist Carbonic Acid. Summary of Results. Reference number. X. U. A. T. Tempera- ture. Gas pressure. Ionic velocity. Negative. Positive. 1 2 3 4 5 6 7 8 9 10 11 302 3-02 302 3-02 6-07 6.07 607 607 6-07 6-07 6-07 8-48 8-52 16-6 16-75 9-74 9-97 19-4 181 8-63 1305 12-9 + 1916 -21-1 -39-2 + 35-9 -12-3 + 11 + 21-5 -22-14 + 9-95 + 14-1 - 15-85 -36 -36 •18 ■18 -62 -61 •31 •33 •70 •47 •47 °C. 163 163 166 16-6 16^9 16-9 17^1 171 17^1 176 176 756 75-6 76-2 76^2 75 75 75-5 75-5 75 75^1 75^1 •679 •717 •658 •685 •678 •745 •791 •755 -756 •722 •772 ^^ ^ P 73/ f/V e. — "^ ■^ . » . . -— -* X ^^ ^ — — ■ "*" X ^^ ^ — ■ , — ^ ' — "^ ft- ^^ — - "^ a— ...-. — — Ni J?^ th e. "^ •1 • i 7 X \ .i 1 4 ^ .« ./ » mi f ^i i 4^ J T in seconds. Fig. 14. The results are represented in fig. 14, from which it is seen that the values corresponding to T = 0, when corrected similarly to those of moist oxygen, give as the velocity in moist carbonic acid, for the negative ions, 75 centim. per second, and Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 225 for the positive ions, '825 centim. per second, for a pressure of 76 centims. and a temperature of about 17° C. A summary of the results obtained for dry carbonic acid is given in Table XVI. Table XVI. — Dry Carbonic Acid. Summary of Results. Beference number. X. U. A. T. Tempera- ture. Gas pressure. Ionic Velocity. Negative. Positive. 1 2 3 4 5 6 7 8 9 10 11 12 607 607 607 607 308 308 308 308 601 6-01 6-01 601 8-61 8-63 16-7 171 17-3 17-3 8-25 8-53 8-63 8-53 12-8 12-85 - 915 + 9-59 + 19-07 - 18-05 -36-2 + 38-4 + 18-96 -18-18 - 9-53 + 9-64 + 14-76 - 1404 •71 •71 •36 •36 •18 •18 •37 •36 •70 •71 •47 •47 -c. 17-5 17-5 175 175 18-3 183 17-2 17-2 173 173 17-5 17-5 75-4 75-4 75-8 75-8 75-8 75-8 75-7 75-7 75-7 75^7 75-9 75-9 -787 •796 •793 •781 •770 •777 •752 •737 •747 •725 •753 •738 The results are represented in fig. 15. The velocities appear to vary but little with T. The values for the positive velocity being comparatively large for the highest value of T, make it difficult to draw the line through the positive points, and the inclination of the one through the negative points has been used as a guide for drawing the one shown. The value thus found, when corrected, gives the velocity in dry carbonic acid for the negative ions as '81 centim. per second, and for the positive ions as 76 centim. per second for a pressure of 76 centims. and a temperature of 17*5° C. i N€ S/K ?. =sss — ^ — -<t = °-" _a ^^^ ^ — = X -F\ ^Si Wi f. y% J . d f — .€ \ 4 1 ,4 ^ •% 1 \ — •i t s VOL. CXCV. — A. T in seconds. Fig. 15. 2 G Digitized by Google 226 MR. J. ZELENY ON THE VELOCITY OF THE IONS § 11. Hydrogen. The gas was prepared from pure zinc and hydrochloric acid, and bubbled through three bottles of strong caustic potash and potassium permanganate to free it from the acid and other impurities. Great difl&culty was experienced in maintaining the gas sufficiently pure on standing, because of the large surface of rubber exposed in the gas bag and in the connecting tubes of the apparatus. The density of hydrogen being so small compared to that of air, a small amount of the latter produces a large change in the density of the gas, and it was found that the ionic velocities were greatly affected thereby. The following plan was finally adopted as the most practicable under the circum- stances : — The forenoon of a day was spent in the preparation of fresh hydrogen, this length of time being required to generate the large quantity necessary for use and for washing the more impure hydrogen out of the apparatus. Beginning early in the afternoon, readings were taken as rapidly as possible until after midnight, thus giving about eleven hours of continuous observations. The density of the gas was then determined by weighing a 600 cubic centims. flask filled first with dry air and then with dry gas from the gasometer. Since 1 per cent, of air in the gas made a difference of over 6 milligrams in the weight, this permitted a sufficiently accurate determination of the amount of the air impurity. A test was made by the eudiometer method, which showed that the impurity was practically all air. The width of the beam of rays used was '3 centim., as the conductivity was much less with the hydrogen than in the other cases. The following is a set of readings taken for the negative ions in dry hydrogen : — Temperature = 20° C. X = 2*95 centims. Barometer = 76*15 centims. Excess pressure in gasometer = '90 centim. „ ,. apparatus = 'SG centim. 5 cells = 10*5 volts. Table XVII.* — Dry Hydrogen. Negative Ions. Voltage of outer cylinder. Electrometer deflection Descent of gasometer in 30 seconds. in 40 seconds. Cells. Diviiiouii. Centinis. -2-6 12-5 9-59 -3 9-7 9-52 -3-4 6-5 9-31 -3-8 3-8 9-44 -3-6 6 9-46 -3-2 . 8-8 9-50 -2-8 11-2 9-38 -2-4 14-2 9-40 Digitized by Google PRODUCED IN GASES BY EONTGEN RAYS. 227 The results are shown graphically in curve II. of fig. 16. The corrected value of 17= 35*0 centims. per second. A = 9-04 volts. V = 5-118 35 2-95 X 9-04 pressure becomes 676 centims. per second. 9-Q5 T = ^ - = -084 second. The gas in this case contained 3 '4 per cent, of air. = 672 centims. per second, which reduced to 76 centims. L f n J / lA 7 f J F^ t / lO 7 y % L / A 1 / ^ t 7 ^»^ '4 M / O t t f -J 7 7 A 4± ^ ( 7 i^"' 7 f J t / -lV : : ( ^. i u 4t. VoLta^ in ceUs. Fig. 16. The following set of readings was taken for the positive ions in hydrogen saturated with aqueous vapour : — Temperature = 20° C. X = 2*95 centims. Barometer = 767 centima Excess pressure in gasometer = '78 centim. „ „ apparatus = '35 centim. 9 cells =18-5 volts. 2 G 2 Digitized by VnOOQ iC 228 MR. J. ZELENY ON THE VELOCITY OF THE IONS Table XVIII. — Moist Hydrogen. Positive Ions. Voltage of outer Electrometer deflection Descent of gasometer cylinder. in 30 seconds. in 40 seconds. Cell.. DiTisiona. Centims. + 6-6 5-25 10-38 + 6 7-75 10-23 + 5-4 10-25 10-18 + 4-6 14 10-38 + 5-4 10-75 10-25 + 6 7-75 10-18 + 6-6 5 10-16 The results are shown in curve I. of fig. 16. The corrected value of U = 43 '3 centims. per second. A = 15-9 volts. 43-3 i; = 5-118 <^n^ i crn = 473 centims. per second, which reduced to 76 centims. 2-95 X 15-9 ^ pressure, = 4'80 centims. per second. 2*95 T = ^ = '068 second. Besides the water vapour, the gas in this case contained 1 "5 per cent, of air. A summary of the results obtained with dry hydrogen containing 3*4 per cent, of air is given in Table XIX. On account of the smaller electrometer readings the Table XIX. — Dry Hydrogen. Summary of Results. Beference number. X. U. A. T. Tempera- ture. Gas pressure. Ionic velocity. Negative. Positive. 1 2 3 4 5 fi 7 8 9 10 11 12 13 14 15 16 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 2-95 21-2 21-6 21-8 21-9 21-6 21-4 21-4 21-4 35-4 35-3 35-0 35-0 34-9 34-7 25-7 25-7 + 6-65 + 7-73 + 7-14 + 7-37 - 5-79 - 5-75 - 5-85 + 7-14 + 10-35 + 10-79 - 9-03 - 9-24 - 8-90 + 11-10 + 8-08 + 7-90 -083 -083 -084 •084 -084 -085 -115 -115 •u. 21-4 21-4 21-4 21-4 21-4 21-4 21-4 21-4 20 20 20 20 20 20 20 20 76-3 76-3 ' 76-3 ' 76-3 76-3 76-3 76-3 76-3 76-5 76-5 76-5 76-5 76-5 76-5 76 4 76-4 6-52 6-49 6-39 6-77 6-eo 6-84 5-56 5-15 5-33 5-18 5-24 5-97 5-71 5-45 5-55 5-67 Digitized by Google PKODUCED IN GASES BY RONTGEN RAYS. 229 determinations for hydrogen are less accurate, and so results were obtained for small values of T, only because of their greater importance, and in order to expedite the readings. The results are represented by I. and II. of fig. 17. The lines projected to T = indicate for the uncorrected velocity of the negative ions 7 '3 centims. per second, and for the positive ions 6*2 centims. per second when under a pressure of 76 centims., and at a temperature of about 20° C. These values are for dry hydrogen containing 3*4 per cent, of air. The correction for the presence of the air can be found approximately by finding the value of the velocity in a gas having a larger percentage of air. z Ni 5^ th e. y X' /. .^ y 7. O J / o y ^ i i °« iCi ve. y «0 — 3r y X' « 2j, X X ^ • t^ M y X f ^ » n. •5 w -^ o X ^ ^ ''^ ^ ^ -^ ^* X n, ^ »l \ • ♦-2 x> • T in seconds. Fig. 17. ./ The following are a number of results obtained with dry hydrogen which contained 14*4 per cent, of air : — Table XX. — Dry Hydrogen with 14*4 per cent, of Air. Reference number. X. U. A. T. Tempera- ture. Gas pr/)ssure. Ionic velocity. Nogative. Positive. 1 2 3 4 2-95 2-95 2-95 2-95 20-4 20-2 18-6 18-6 + 9-62 -8-28 -7-59 + 8-90 •15 •15 •16 •16 "C. 21-2 21-2 ■ 22 22 76-3 76-3 76-4 76-4 4-25 4-27 3-70 3-65 Digitized by VjOOQ IC 230 MR. J. ZELENY ON THE VELOCITY OF THE IONS These results are represented at III. in fig. 17. By finding the difference between these points and the values in the curves above them corresponding to the same value of T, the diminution in the velocity is obtained that is produced by the addition of 14*4 — 3 4 = 11 per cent, of air. Assuming that up to this point the diminution in the ionic velocity is proportional to the amount of air present in the gas, the velocity in pure hydrogen is found by adding to the value 34 obtained when 3 '4 per cent, of air was present tj part of the diminution observed as due to 11 per cent, of air. From the above results this correction is found to be '65 for the negative ions and '50 for the positive ions. Disregarding any minor correc- tions, the final result for pure dry hydrogen is thus found to be 7 '9 5 centims. per second for the negative ions, and 6*70 centims. per second for the positive ions at a pressure of 76 centims., and at a temperature of about 20° C. A summary of the results obtained with hydrogen saturated with aqueous vaj)our, and containing 1 '5 per cent, of air, is given in Table XXI. . Table XXI.- —Moist Hydrogen. Summary of Results. Ionic velocity. Reference number. X. U. A. T. Tempera- ture. Gas pressure. Negative. Positive. I 1 2-95 43-9 -U-6 •067 20 771 526 2 2-95 43-7 -15-0 •067 1 20 771 510 — 3 2-95 43-7 + 15-5 •067 20 77-1 — 4-97 4 2-95 43-3 + 15-9 •068 20 771 — 4-80 5 2-95 431 -150 •069 20 771 503 — 6 2-95 23-8 - 8-43 •12 19^8 .76^9 4^98 — 7 2-95 23-5 - 8-76 •13 19^8 76^9 4-72 — 8 2-95 23-4 - 8-64 •13 19^8 76^9 4^77 — 9 2-95 23-3 + 8-77 •13 19^8 76-9 4-68 10 2-95 23-3 + 8-69 •13 198 76^9 4-73 11 2-95 I 34-2 + 13-3 •087 19^8 77 4-53 12 2-95 341 + 13-8 •087 19^8 77 4-37 i 13 2-95 34 -12-4 •087 198 77 4-82 — 14 2-98 19-8 - 8-74 •15 204 76-7 3-93 — 15 2-98 19-7 + 8-97 •15 20-7 76^9 ■~~" 382 air The results 1 to 13 are represented by IV. of fig. 17. The resuJt^ 14 and 15 were obtained with moist hydrogen containing 8 per cent, of ThesQ itwp were selected out of a number of results of which they represent about the ,averag€ values. They are shown by V. of fig. 17, and by means of them the correction for the air present in the above experiments was made in the same manner as with dry hydrogen. The points IV. in the figure are so scattered that the inclination of the lines drawn through them had to be estimated mainly by comparison with those for dry hydrogen, remembering that with the smaller Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 231 velocities here obtained the inclination would be somewhat less. The final values thus obtained for hydrogen saturated with aqueous vapour when corrected for the air present give for the velocity of the negative ions 5 '6 centims. per ^second, and for the positive ions 5*3 centims. per second at a pressure of 76 centims., and a temperature of 20° C. § 12. Eemarks on the Experiments. The changes in the values obtained for the velocity with changes of T are observed to be greater for those cases where the ionic velocities are higher. With dry and moist carbonic acid, however, the inclination of the ciu*ves is some- what different for nearly equal values of the velocities. In some instances, where the set of points for either the positive or the negative ions did not allow of a suflSciently accurate estimate of the inclination of the line to be drawn through them, the line through the other set of points was used as a guida The presence of water vapour diminished the velocity of the negative ions in all of the gases, while in carbonic acid the velocity of the positive ions was at the same time considerably increased. It seems most probable that these changes are due to some effect upon the size of the ions, and it is possible that a few molecules of the aqueous vapour collect upon the negative ions. It is interesting to note in this con- nection the recent results of C. T. R. Wilson,* showing that in supersaturated air the water condenses more readily upon the negatively charged ions. While in most cases the readings indicate a greater accuracy, it is believed that the maximum error in any determination is less than five per cent. For convenience, all of the values obtained are here collected in one table, the results being given in centims. per second both for a field of one volt per centim. and for a field of one electrostatic unit per centim. Table XXII. — Ionic Velocities. Gas. Velocities in centims. per second in a field of 1 volt per centim. Velocities in centims. per second in a field of 1 E.S.U. per centim. Eatio of Negative to Positive. Tempera- »ture. Positive ions. Negative ions. Positive ions. Negative ions. Air, dry „ .moist Oxygen, dry .... „ moist . . . Carbonic acid, dry . . „ „ moist . Hydrogen, dry . . . „ moist . . 1-36 1-37 1-36 1-29 •76 •82 670 5-30 1-87 r5i 1-80 1^52 •81 •75 7^95 5-60 408 411 408 387 228 246 2010 1590 561 453 540 456 243 225 2385 1680 1-375 MO 1-32 1^18 107 •915 119 1-05 "C. 13-5 14 17 16 175 17 20 20 * C. T. R. Wilson, *Phil. Trans.,' A, vol. 193, p. 289, 1899. Digitized by Google 232 MK. J. ZELENY ON THE VELOCITY OF THE IONS It is seen that the value of the velocity is greater for the negative ions in all cases except for moist carbonic acid. In comparing the values for the diflferent gases, the temperature at which the observations were taken must be taken into consideration. At the time the writer* determined the ratio of these velocities, the influence of moisture being unknown, the gases used were not dried, and so the values obtained were between those given above for the dry and the moist gases. Of the gases used in the former experiments, which were not used in these, the ammonia gas used had been passed through two long tubes of calcium oxide, the acetylene gas had been passed through a long tube of calcium carbide and the nitrogen monoxide was used directly from a cylinder. The results obtained by E. RuTHERFORDt for the sum of the velocities of the ions produced by Rontgen rays are for : — Air = 3'2 centims. per second. Oxygen =2*8 centims. per second. Carbonic acid = 2*15 centims. per second. Hydrogen = 10*4 centims. per second. It is not stated whether the gases were dried, but the value for air agrees with that given above for the sum of the velocities in dry air, while the values for oxygen and hydrogen agree with the values for the moist gases. The value for carbonic acid is nearly 40 per cent, larger than that obtained here. It is of interest to compare the velocities of the ions produced by Rontgen rays with those of the ions produced by the action of ultra-violet light and in the discharge from points, as they show a close similarity. For conduction produced by ultra-violet light, E. Rutherford:|; obtained with dry gases for the velocity of the negative ions in — Air = 1'4 centims. per second. Hydrogen = 3*9 centims. per second. Carbonic acid = '78 centim. per second. The value for carbonic acid is quite near to that obtained above, but the other two are considerably smaller. A. P. Chattock§ obtained for the velocities of the ions in dry air in the case of discharge from points — 413 centims. per second for the positive ions, and 540 „ „ „ negative ions for a field of one electrostatic unit. * J. Zeleny, *Phil. Mag.,' July, 1898. t E. Rutherford, ' Phil. Mag.,' November, 1897. t E. RuTHERiX)iU), ' Proc. Camb. Phil. Soc.,' vol. 9, Pt. VIII. § A. P. Chattock, *Phil. Mag.,' November, 1899. Digitized by Google PRODUCED IN GASES BY RONTGEN RAYS. 233 These values are nearly the same as those obtained above for the ions produced by Rontgen rays. J. S. TowNSEND* has shown that from the ionic velocity in a gas and the coeffi- cient of diffusion of the ions in the gas, the value of Ne can be obtained, N being the niunber of molecules in a cubic centim. of the gas, and e the charge carried by an ion. By comparing this value with that obtained from the electrolysis of liquids, the relation between the charges on the ions in the two cases can be determined. Using the values of the ionic velocities (v) given in Table XXII., and the corre- sponding coefficients of diffiision (K) from the tables given by J. S. Townsekd, the ^ V 1 ft® 1) values of Ne are obtained from the equation Ne = — = for the positive and the negative ions in both dry and moist gases. The results are given in the following table : — Table XXIIL— Values of Ne X lO'i^. Gas. Moist gas. Dry gas. Positive ions. Negative ions. Positive ions. Negative ions. Air 1-28 1-34 1-24 101 1-29 1-27 M8 •87 1-46 1-63 1'63 1-31 1-36 1-25 Oxvfiren Hvdrofifen Carbonic acid •99 •93 The corresponding value of Ne obtained for hydrogen from the electrolysis of liquids is 1*23 X 10^^ at a pressure of 76 centims. of mercury, and a temperatiu*e ofl5°C. The values of Ne in the table for the positive and the negative ions in moist air, oxygen and hydrogen are perhaps in sufl&cient agreement to justify the statement that the charges carried by the positive and negative ions are the same, and that the value is also the same for the three gases, and corresponds to the charge carried by the hydrogen ion in the electrolysis of liquids. The values of Ne for the negative ions in the same three gases when dry are not far from those in the moist gases, but the results for the positive ions are consider- ably larger. It seems very improbable, however, that the charges carried by the ions are different in the moist and dry gases, since most likely the moisture does not in- fluence the act of the ionization itself, but either affects the ions after they are formed during the production of clusters of molecules around them, or changes the resistance VOL. CXCV. — Jl, * J. S. TowNSEND, *Phil. Trans.,' A, vol. 193, p. 152. 2 H Digitized by Google 234 VELOCITY OP IONS IN GASES. to their motion. So if the charges are equal in the moist gases, they should be equal in the dry gases also. The values of Ne for carbonic acid are all less than that obtained for hydrogen by electrolysis, and so indicate a smaller charge on the ions ; but from analogy with liquids we should expect that if the charges vary at all, it would be in the ratio of one to two or more, unless it is possible to have a charge smaller than that carried by hydrogen in electrolysis. The writer cannot account for the differences in the values of Ne by supposing them due to errors in the ionic velocities obtained, since that would mean the pre- sence in the experiments of some error which in some cases influenced the results for the positive ions alone, in other cases had an effect upon the values of both of the ions, and in still other cases was without effect. The experiments described in this paper were performed at the Cavendish Labo- ratory, Cambridge, and I desire to express here my thanks to Professor J. J. Thom- son for the encouragement and valuable suggestions given in the course of the investigation. Digitized by Google [ 235 ] VI. Undergrmind Temperature at Oxford in tJie Year 1899, as determined by Jive Platinum-resistance Thermometers. By Arthur A. Eambaut, M,A., D.Sc, Radcliffe Observer. Communicated by E. H. Griffiths, F.RS. Received May 17,— Read June 21, 1900. [Plates 1, 2.] Description of the Apparatus and Mode of Reduction of the Observations of Earth Temperatures. The instruments with which the earth-temperatures given in this paper were observed, were five platinum-resistance thermometers of the Callendar and Griffiths pattern,* made by the Cambridge Scientific Instrument Company. These were purchased by the late Mr. Stone, and were placed in position under his direction shortly before his death. The method of platinum thermometry seemed to be particularly suitable for this class of work, on account of the immunity it enjoys from certain errors attending the use of the long-stemmed mercurial or spirit thermometers ordinarily employed for underground temperatures. A higher degree of accuracy might, therefore, reasonably be expected, and the discussion which follows of the first complete year's observations at the Radcliffe Observatory shows, I think, that this anticipation has been justified. Some discrepancies between theory and observation no doubt appear, but they are of a character which seems to indicate a difference between the assumptions on which the theory is based and the conditions actually prevailing in the stratum of gravel in which the thermometers are buried, rather than thermometric errors affecting the observations themselves. The thermometers are inserted in undisturbed gravel, the first four lying one under the other, in a vertical plane beneath the grass of the south lawn, and within a few feet of the Stevenson screen in which the dry and wet bulb, and the maximum and minimum, thermometers are suspended. In order that the thermometers might lie in practically unbroken ground, the following method of placing them was adopted. A pit was dug at the edge of the * See the Cambridge Scientific Instrument Ck)mpany's " Descriptive List of Instruments,^' page 20. 2 H 2 22.11.1900 Digitized by Google 236 DE. A. A. RAMBAUT ON XJNDERGROUND TEMPERATURE AT OXFORD grass about 5 feet long by 4 feet wide. One edge of the pit coincided with the edge of the grass plot, and the corresponding side of the pit was made as nearly vertical as possible. Into this vertical face four iron tubes were driven horizontally, the tubes being formed with spikes at their ends to facilitate this operation. The tubes are 4 feet long, and into them the thermometers were inserted with the leads attached, the mouths of the tubes were sealed up with tow and red lead, and the pit filled in. The first four thermometers were placed at depths of (approximately) 6 inches, 1 foot G inches, 3 feet 6 inches, and 6 feet respectively ; but Mr. Stone soon saw the advisability of placing another at a lower level, and intended to have gone to a depth of 20 feet. But as water was met with at a depth of 10 feet 6 inches, he decided to place it just above the water level, at a depth of 10 feet. This thermometer was buried, not directly under the four earlier ones, but in a separate pit at the other side of the Stevenson screen. This was apparently done to avoid disturbing the leads of the thermometers which were already in position, but it would have been rather more satisfactory if all had been placed in the same vertical plane. It is also, perhaps, to be regretted that one or two similar thermometers were not buried to considerably greater depths. The presence of water, however, complicated matters and introduced conditions different from those which prevailed in the dry gravel above. It is not, for example, to be supposed that the thermal conductivity or the difiiisivity of permanently water-logged gravel would be the same as that of the drier material above it. Hence it would appear necessary to put at least two thermometers below the permanent water-level in order to study the flow of heat under such circumstances. Besides, it is highly probable that the gravel stratum is not very much thicker than. 10 feet. Excavations in the neighbourhood show that the blue Oxford clay is likely to be met with at any depth below 12 feet from the surface, and in this, of course, the thermal conditions would be likely to prove wholly different from those in the gravel. The actual depths of the various thermometers as measured in October, 1898 (when the pits were standing open to enable us to re-standardise the thermometers) were as follows : — Thermometer .12 3 4 5 Depth ... 6^ in. 1 ft. 6 in. 3 ft. 6^ in. 5 ft. 8^ in. 9 ft. 1 1| in. These thermometei's, with the Callendai* and GriflEiths resistance box, which could be connected with each thermometer through a switchboard, had been set up as I have stated, shortly before Mr. Stone's death. On my appointment to the post of Radcliffe Observer, I took an early opportunity of examining the apparatus, and partly with a view of familiarising myself with all its details, I proceeded to determine the comparative values of the coils, and to Digitized by Google AS DETERMINED BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 237 re-standardise a spare thermometer which was kept in the observing room for general purposes. This examination led to the discovery of discrepancies in the readings of the apparatus which troubled me for a long time, and which necessitated a large number of experiments extending at intervals over the greater part of a year before they were traced to their sources and eliminated. In this part of the work I have to acknowledge the very generous help and advice of Mr. E. H. Griffiths, F.RS., who was kind enough to come to Oxford on more than one occasion to place his experience at our disposal, and who, at one stage of the investigation, took the resistance box and spare thermometer to Cambridge to subject them to a prolonged examination in his own laboratory. These discrepancies, though serious in view of the accuracy which we had reason to expect from the apparatus, were still small quantities confined within one or two tenths of a centigrade degree. They were, for the most part, traced eventually to uncertainties in the contacts at the switchboard, and a want of perfect insulation in the older leads. These consisted of four india-rubber covered wires which, in the underground portion, passed through leaden pipes, but within the observing room were without the leaden covering. It was found that these were very susceptible to damp, and that the insulation fell away very rapidly when there was much moisture in the air, thus giving rise to very puzzling and troublesome discrepancies. In September, 1898, the switchboard was improved and new composition cable leads substituted, which extended without interruption from the thermometers right up to the switchboard. Since these changes were eflfected the discrepancies have ceased to appear, except on one occasion (viz., October 27, 1899), when it was found that the short flexible lead from the switchboard to the resistance box was thoroughly damp. On lighting a fire in the observing room to dry the covering of this lead, the irregularities disappeared. Since that date up till the end of March of this year (1900) I have kept a gas light burning continuously in the room, to prevent the deposition of moisture, and have experienced no ftirther trouble of the sort. The resistance box is in its general design similar to that described by Mr. Griffiths,* but simplified to suit the particular class of work for which it was intended. It is provided with three principal coils A, B, C, whose nominal values are 20, 40 and 80 box units respectively, a box unit being about 01 ohm. There are two additional coils, one for the calibration of the bridge wire, and another, which we have called the " concealed coil," whose value is about 240 box units, which was inserted for convenience to balance approximately the resistance of the thermometers at 0° C. when the coil A was also in the circuit, so that the reading of the bridge wire under these circumstances might be as nearly zero as possible. * ' Nataroi* Noyember 14, 1895. Digitized by Google BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 239 oscope is placed on a window ledge to the right of the position that the observer can manipulate the commutator 1 of the current without removing his eye from the eye- !it of the apparatus is shown in fig. 1. To the right is oroscope ; underneath in front is the commutator, and . On the extreme left is the switchboard, and in the a small electric motor for stirring the oil in which the d. the apparatus the method described by Mr. Griffiths in 1895, was in the main followed. The temperature coeffi- \ Griffiths when the apparatus was under examination Cambridge. Two separate determinations made in 1898 e the following results : — Range of Temp. Temp. Coeff. 27 . . 9°-18 0-000242 } 8 . . 12 -51 0-000240 bservations, the value 0*00024 has been adopted. The been borne out by subsequent observations in several ample, the invariable steadiness in the changes of No. 5, erature of the box, indicated a high degree of accuracy instant. he coil values and the unit of the bridge wire scale, the lade at the Radcliffe Observatory : — )51 B-A= 19-851 A = 19-603 )46 19-849 •600 143 19-848 •601 — 19-853 -602 - 19-847 47 19-850 19-601 Mr. Griffiths* paper referred to above, = 80-158-| = 39 '979 >mean box units. = 19-863 I •idge ^wire is equal to I '0134 mean box units. ible giving the correction for the particular arrange- Digitized by Google AS ICmLJUMT In iLis eTT«ressJ c K» :s the i^ts:st.ii--:>r of :be " vi: that or-II in tbe i^:-L-.i 1 mined fi>:»cn tiie o'teerrh:: OMnbined with the cccst.^:. sponding resistances R a:. Thus, if X be the ralue wire when the themior/.r: Iwidge wire when it is in.^ the total resistances in t:. ratio of these resist^LCc^ used in the oonstructioa 03872 The values of X found and 6, 1898, for the purpc* Thermci 1 4 5 A For any arrangement of the total resistance in the correction for temperature We thus find the folic been used in the observat VOL. cxcv. — A, Digitized by VnOOQ iC rtVE PLATINUM^RESISTANCE THERMOMETERS. 241 ^efficient, 0*00024, as determined by Mr. Griffiths, .^ = R, X 0-00024 X (^ - 14°). total resistance in the circuit, and since this includes id coil," we require to know approximately the value of 3 of the equation. This is, perhaps, most easily deter- )f the thermometers themselves at 100° C. and 0° C, blue found by Mr. Griffiths for the ratio of the corre- his coil, Tq that of the other coils in use and the bridge s packed in melting ice, and r^ that of the coils and 3d in steam, reduced to mean box units at 14° C, then .vo cases are, X + ^'i and X + ^o> and if we take the be 1*3872,* as found by Mr. Griffiths for the wire X + r this instrument, then rr— — - = 1'3872, and therefore ^ + ^0 this way from the observations made on October 4, 5, )f standardising the thermometers, are as follows : — jr. X. 240-65 -65 '77 77 -60 -65 Mean 240-68 lis (Y) and any bridge wire reading (R) we have therefore cuit, X + Y + R> and the coefficient of {d — 14°) in the (X + Y + R) X 0-00024. ng table for the two different arrangements which liave s: — 'Nature,' November 14, 1895, p. 45. 2 I Digitized by VnOOQ iC AS DETERMINED Bl Thermometer . 1. 1898, October 4 5 101-286 Adopted values , 101-29 To those who have standa the separate results for the however, he pointed out t while they were sealed up attached ; hence it was imj along these tuhes, but for re take further precaution agai mining the zero points the any error arising from this c One of the most import? degree of permanence in i intervals of time. It was tl found in my first observe previously at the time that all the thermometers exhni apparatus. This examine ti insulation of which was fou Another series of discrej the switchboard. In the ( lead firom the resistance Ik also were led the four bn By having the steel pron finnly pressed against the 1 was experienced ; and, sine the same cause. It has b time with the four steel i check on the character of t Taking advantage of a thermometer No. 1 (6 inch a year 8 continuous obser year, the zero point of thi less than 0°005 C, the ac I] andii Digitized by VnOOQ iC FIVE PLATINUM-RESISTANCE THERMOMETEES. 243 Temperature of Steam. 2. 3. 4. 6, 101-474 101-610 -589 101-389 -367 101-179 101-47 101-60 101-38 10118 Used naked platinum thermometers the discrepancies in 3. p/s of Nos. 3 and 4 may appear larga It should, it it was necessary to standardise these instruments n strong brass tubes with heavy leaden-covered leads •ssible to eliminate altogether the effects of conduction sons given on p. 244 it was not considered necessary to st the small errors arising from this cause. In deter- thermometers were placed in a trough 3 feet long, and ,use was very much diminished. it considerations in connection with this subject is the le fundamental points as determined at considerable 3 occurrence of discrepancies between the values which I dons and those determined about a year and a half he instruments were set up, which induced me to have 3d and to make a thorough re-examination of the whole n led eventually to my discarding the original leads, the d to fall off very much when they became damp, mcies was traced to an uncertainty in the contacts at [•iginal form the four steel prongs in which the fourfold c terminates, were inserted into mercury cups into which 58 strips to which the thermometer leads were soldered. s amalgamated, and adding springs to keep each prong rass strips immersed in the mercury, a great improvement I this change was effected, we have had no trouble from 3n the habit, too, to make the observations from time to ongs in both positions, which affords a very satisfactory e contacts. visit from Mr. Griffiths on October 6, 1899, I had s) dug up, and we examined its zero point after exactly ations. Determined in the same way as in the previous thermometer was found to agree with the earlier value to lal values being 1898 0'806 1899 0'802 2 I 2 Digitized by Google AS DETERMINEE A complete determina of the sunken thermomet (1) The balancing of and the temper; (2) To R is to be adde from Table I. (3) The correction to Table XL (4) The reduction to Table III. mult (5) The correction froi It only remains to re the air scale. The relation connectin in which pt is the platii a constant. For a completely inde the resistance at some but the experiments of ' value of 8 varies from on particular sample of wii are given in Mr. Griffit The value of 8 for th mined at Cambridge t( apparatus for the deter doubtless have been adv this constant. Since, Ik variations of earth tern within that range the cci 0*050 (which is quite m Writing ^« + dforti 0'003, its square may be T being written £cvptil( * 'PhiL Trans./ A, 18S7. t Cy. the Report of the C Practical Standards for use in Digitized by Google ' FIVE PLATINUM-EESISTANCE THEEMOMETEES. 245 of temperature on the platiuum scale by means of one is, therefore, reduced to the following simple steps : — galvanometer and reading of the bridge wire scale (R) L'e {0) of the box. le correction for the particular arrangement of coils used, luce the bridge wire reading to mean box units, from indard temperature (14°). The quantity taken from ed by {0 —14) gives this correction, ^able IV. 3e the temperature thus expressed from the platinum to bese two, established by Professor Callendar,* is -'-p"ii^)'-U w 1 temperature, t the temperature on the air scale, and 8 ident standardisation it would be necessary to determine rd known temperature in order to obtain the value of 8, LLENDAR and Griffiths have shown that although the ipecimen of platinum to another, it is a constant for any References to the original papers bearing on this point * article in ' Nature ' cited above. )articular wire used in the Oxford instrument was deter- )e l'512.t If it were intended to employ the Oxford Qation of temperatures over a very wide range, it would ble to make an independent determination of the value of 3ver, the range —15° C. to +25° C. will cover all the matures with which alone we are here concerned, and since iction does not amount to as much as 0'3, an error of even lissible) in the value of 8 would not affect our results, equation (6), and remarking that since dj 100 is less than jglected, we find = 8(t^ - t)/{1 + (1 - 2r)8/l00} hiittee of the British Association for improving the Construction of metrical Measurements. Bradford, 1900. [September 16, 1900.] Digitized by VnOOQ iC X-^ iS Ulkf ^1^ rr^ fc-rc :z. -lijt . results. ATii Jis tie : f^r disdiict adv^LHt^o? Ir. rv: On aocoant of the ir.'i carded them alt* -gether, pcfftioDS which are alte: : vations are taken only oi of days in each divisir. lengths as small as {k^^ thirty and thirty-one c intercalating the extra c ♦ Professor W. Thomsox, Eoy. Soc. Edin./ voL 22, p. t ' Greenwich Observatior t Professor Everett, ' T: Digitized by Google LMIMD BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 247 . . 9-550." Sept. 22 . . 14-42 C. . . 9-59 . . 9-63 •CoaA. 23 . 24 . . 14-41 14-41 . . 9-66 25 . 14-40 . . 9-70 26 . . 14-39 . . 9-75 . . 9-80 •CoaR 27 . 28 . . 14-37 . 14-35 , . 9-82 . 29 . . 14-34 SCoilR I Coil A. Discussion of the Obso^^atiojis. the discussion of the observations is to group them into monthly ) to deduce the harmonic expressions which will represent the ermometer throughout the year.* be work I have adopted the Fahrenheit scale, as the observations educed to this scale for comparison with our other meteorological observations of the same kind at Greenwicht and Edinburgh| sor Everett are expressed in the same scale, there seemed to be a in retaining it, e inequality in the lengths of the calendar months I have dis- bher, and, as far as possible, have divided the year into twelve iltemately thirty and thirty-one days in length. As the obser- ily once a day, it is of course necessary to have an integer number vrision, but the following scheme makes the differences in their I possible, and with one exception, that of January, alternately ►ne days. In Leap Year this exception would be removed by bra day in January, instead of February. 3N, " On the Reduction of Observations of Underground Temperature," * Trans. p. 409. ^ions/ 1860 (cxciii.). Trans. Roy. Soc. Edin.,' vol. 22, p, 429. Digitized by VnOOQ iC AS DETERMINE] McClellan in the or Mr. WiCKHAM or Mr. and experience, and, as able degree of precision. These means were d any modification or cor to above, except on one flexible lead was found room. This was indicated b 10 feet thermometer, \ steadily, that its readin one-twentieth of a degr drying the lead, howeve the subsequent readings as before the diBcrepancj As the dampness of and in no way affected take an interpolated vali the difference between added as a correction to j This particular case i] deep sunk thermometer apparatua The monthly means an periods of two months ea been selected, as the fir the 10 feet thermometer in the indications of this amplitude of a wave is thermometer to the one 1 In fig. 3 are given surface, deduced from th( The harmonic expres thermometer throughout e or 0^ao+t where t denotes the time VOL. cxcv. — A. Digitized by Google MINED BY FIVE PLATINUM-RESISTANCE THERMOMETEES. 249 he ordinary routine, but during his vacation, or on Sundays, • Mr. Robinson took his place. All three are observers of skill id, as the results seem to show, the observations are of a remark- iision. ere deduced from the observations as directly obtained without )r correction, other than those taken from the tables referred 1 one date— October 27 — when, as I have mentioned, the short mnd to be affected by the dampness of the air in the observing led by a sudden change of about 0°*13 R in the reading of the er, which, under] ordinary circumstances changes so slowly and 3ading on any day might be predicted with certainty to within degree from the readings of two or three days preceding. On ^ever, the abnormal readings disappeared by the next day, and ings of this thermometer were foimd to lie along the same curve •ancy had arisen. of the lead disturbed only the reading of the resistance box cted the thermometers themselves, we were, therefore, able to value for the reading of No. 5 as a standard of comparison, and )en this and the actually-observed readings, viz. : 0°*13, was to all observations made on that day. je illustrates very well the protection which the readings of a ter afford against sudden changes occurring unobserved in the 1 are graphically represented in fig. 2 ; the daily readings for two 3 each are exhibited in Plates 1 and 2. These two periods have first includes the minimum and the second the maximum of er, and both illustrate very well the steadiness of the changes his instrument, and exhibit also the manner in which both the is dinainished, and its phase retarded in passing from one e below it. n the mean monthly temperature gradients beneath the :he same figures. ression to represent the temperature of any particular it the year will be ^ = aQ + a^ cos \t 4" % ^^ 2X« + &c. + hi sin \t + 62 sin 2X« + &c (c) Pi sin (Xe + El) + P2 sin (2Xt + E^) + &c (d) e represented as the fraction of a year, and X is equal to 2ir. 2 K Digitized by VnOOQ iC AS DETEEMIKEI i o 08 4a u o ^5 »4 I 1 ^'^ ? 'J H 1 1 Digitized by Google •LATINUM-RESISTANCE THERMOMETEES. 251 K) X». «J « K •O .«!> 5f' •399^ Ul tf9c/9Q 2k 2 Digitized by VjOOQ IC DETERMINED are here dealin nuch importanc juracy with ^ the formulae i between then iriBon of Compi Thermometer. January . . . February March. April . May . June . July . August September . October . November . . December . . ; appears that w r the formulae, 1 ^ially in the ca^ coselves felt to a rfece of the gn are sunk, beinj a fairly uniform at will be repres K denotes the ter below the bi ion of this is 'ing this expres Digitized by Google :>LATINUM-^RESISTAITCE THERMOMETEKS. 253 observations of a single year, it would be unsafe 3 deducible from the smaller terms. Dbservations are represented by the first three by the following table, which contains the ily temperatures as computed and those actually bserved Mean Monthly Temperatures, C — O. 2 3 4 ■ ■■ 5 -0°86 - d'-55 - 0°32 0°-00 -0-18 + 0-07 + 015 + 0-05 + 0-71 + 0-48 + 0-22 + 0-02 -0-70 -0-48 -019 000 + 0-91 + 0-25 + 0-01 -0-05 -1-02 -0-62 -0-24 -002 + 1-15 + 0-91 + 0-49 + 006 -110 -0-44 -0-07 + 010 -0-15 -0-69 -0-65 -0-23 + 1-87 + 1-29 + 0-73 + 0-07 -214 -107 -0-31 + 0-18 . + 1-62 + 0-75 + 0-18 -018 iree deeper thermometers are fairly well repre- lonsiderable differences in the two upper ones. L, is largely due to the diiUTial variations which bout 3 feet. le neighbourhood of the spot where the ther- lately level and the gravel being, as far as we for a considerable distance in all directions, the ourier's equation 0/dx^ = d0/dt . . (c) of the gravel, and x denotes the depth of a and 2af,finK = — nX. Vt = -^' le series {d) given oa page 249 we have ' and E, = /8,a! + y. Digitized by VjOOQ IC aiMINE] Values c Them com No. 5 a „ 6 „ B „ i » 4 „ 3 Mear d the re y short uried ir contaii • ^/1^/K I those dividua ote hov from tl possibly h separ ). 5 bei the oth "when gravel" n next f close y^early duced )recisic ally la dedu may p ' whic] of th it at Digitized by VnOOQ iC PLATINUM-RESISTANCE THERMOMETERS. 255 iuced from the Half-yearly Wave. From diminution of amplitude. From retardation of phase. 0U45 •1393 •1306 •1287 •1164 •1033 01236 •1187 •1129 •1092 •1022 •0949 0-1271 01102 of both , . . 0-1187 ed from the readings of No. 1, as they seem too iations to aflford reliable results. This thermo- soil which is of quite a different character from ler thermometers. rom the annual wave are, of course, much more rom the half-yearly wave, and the larger discrep- om the latter are not surprising. It is, however roborate the others, showing, for instance, larger son of Nos. 5 and 4 than from that of any other a smaller value of k for the stratum of gravel about two thermometers, than for the higher strata, or it some distance (9 feet 6 inches) from the vertical were open, no very critical examination of the nt depths was made ; but it is proposed to repair )meters are dug up. of the mean values of x/v/k derived from the ry remarkable (especially in view of the fact that servations of a single year), and seems to indicate servations. found from the diminution of amplitude, as com- he retardation of phase must be traced to some ue to the proximity of the Observatory building, 1 at a distance of 36 feet from the thermometers. )eneath the buildings would in all probability be epth beneath the exposed surface. There would, Digitized by Google 5 4 3 2 values i a givei to any ^ main unc oreticalJ^ i^e may c acted. T to 0°-01 1 )th we h; 1 therefore as the de •^1 = or the half X. = at which 0^-1 F. are deals with [)aucles l)et lave been ancles are r a periodic to he dimi physical ai ource of in lely, thernK :he liquid —A, Digitized by VnOOQ iC PLATINUM-RESISTANCE THEKMOMETERS. 257 ^e, we find 1 9 7 2 = amplitude of half-yearly wave at surface. Df equations (/) we can determine the value of r, P,„ or the depth at which the amplitude of the , on the hypothesis that the conditions prevailing greater depths. 10 invariable layer so long as equations (/) are it an annual variation of 0^'02 F. is less than can , therefore, at which the amplitude of the annual ntents and purposes be considered as invariable. P^ = — 2 (M being the modulus of common + log PoO/M^Wk. 3h the amplitude of the annual wave is reduced rich feet = 66*0 English feet, e, ich feet = 36-0 English feet. al and half-yearly waves are reduced to an I similar way to be 45*3 and 21*4 English feet tions of a single year, and the results accordingly J and observations which, although they are less pected, are greater than one would like to see. to the fact that the temperature variations are as the theory supposes, and as such they might he mean of a number of years, and partly to I the surface. ►nsi<lered by Lord Kelvin in his paper, referred ►rs arising from the uncertainty as to the tem- ; stems of the thermometers used in Professor *> T Digitized by VnOOQ iC c^ui. 3^5 6 7 a : . ! ' '■ i : ' : 1 ■ ' i . > 1 i ; ' ■ i ! j : 1 1 ■ 1 i • f ' r ' 1 — ■- r- r 1 7- J ^*'^'' ^ / i/^ •\ / / V, \ \ // \ \ J^ \ 1 > -^ • a d Digitized by VjOOQ IC Phil. Trans., A, vol. 1.95, PL i. It 17 16 19 SO a^ ' ££ es s* es aa £T ea S9 30 V- 54^ / \ fii* A A^ (^ Si^ _^ / / ff- 5CP . \ ^ V - ^ /- / ^ 7i [,•* s:, , ::s; / r^ ^ r J \^IZl ^.. •7 y ^ / £ r ^^-" ^2£S^ ^ X / \ / n^ f— \ / 44^ / \ / 4^ / / 4.2^ 1 / t — 3flP V9^ #■ k90 37^ £ 0yD, •/) /(7i H- %A^ -^■a ^m^. _^ / /if. 6 in. 00 34^ ..— .... .. 3 /3f. ^i in. ..••• .••• 6 Z"^. ^ in. 3J^ _^ ^ ft. Hi in. 1^ 3/ 2^ 5 / 7 1 a i 9 £ 'JO i. <i & 2 k >3 J •'^^ !5 i » i 17 £ a i »« P^ Digitized by VjOOQ IC Phil. Trans,, A, vol. 1.95, PL i. /d 17 18 i9 20 Sil 22 as B^ £5 25 27 as B9 30 9 /7 /a /9 20 2/ 22 23 2^ 25 26 27 26 293C Digitized by VnOOQ iC VIL The Diffusion of lous ^> stance J i By John S. Townsend, M Fellow of Conimunicato Rect A GENERAL method of findiii described in a previous paper,^ with ions produced by Rontge with ions produced by a radi( violet light. Tlie principle of from observations on the los tubing. The experiments were arran greater than the loss due to t> effects which must be consider< binatlon which occurs when tli gas ; and the effect due to the most of the ions are chargt-'f necessary either to correct for the experiments so that the lev The present paper is divid investigation of the relative ii and mutual repulsion in causii and the results of the experinn by ultra-violet light, and by j) respectively. The conclusion^ Section V. In the previous paper we h\ distributed throughout a gas, * JuHN S. Tow Digitized by VnOOQ iC [ 259 ] id in Air by the Action of a Radio-active Sub- violet Light and Point Discharges. hrk'Maxwell Student, Cavendish Lahordiory, / College, Caml/iidge. ro/essor J. J. Thomson, F.Ii.S. J 17,— Rend June 14, 1900. . rate of diffusion of ions into a gas has been 1 account was there given of the results obtained The present paper gives the results obtained substance, by point discharges, and by ultra- tliod consists in calculating the rate of diffusion nductivity of a gas as it passes along metal }hat the loss due to diffusion should be much uses. In order to ensure this, there are two iiig the dimensions of the tubing : the recom- x)th positive and negative ions present in the repulsion of the ions which takes place when electricity of the same sign. It is therefore Lirces of error or to arrange the conditions of ductivity due to these causes is negligible, five sections. The first section contains an e of the processes of diffusion, recombination, conductivity. The descriptions of apparatus, B on ions produced by a radio-active substance, larges, are given in Sections II., III., and IV. l'a^vn from the experiments are discussed in CTION I. that when a number of ions. A, are uniformly is entering metal tubing, the ratio R, of the \. Trans-,' A, vol. 193, 1899, p. 129. L 2 8.12.1900 Digitized by Google BY THE ACTtn positive and negative ions ai»| experiments on diffusion, we a very closely those produced n; therefore assume that the ia\\ • in the two causes. Tlie method explained in the previous pa{M the loss due to reconibinatH»n sides. The time, Zi/V, in the ex\)t second, the radius of the tuhi finer tubing (a = -5 millini.). reduced to ^. The numl)er, N to N/9. The radio-active sulistance the radiation proceeding fron smallest that was used in t therefore assume that in tht» not affect the value of y to th When a gas contains ions i of ultra-violet light on a nieti arising from the electric densi It would be difficult to find tl of ions in a tube while dift\isi( to the error it introduces. Let us consider the c<\Ke ol* owing to the motion of the io we suppose that no diffusion electrification at any jx)iiit is Pq being the initial density, s by unit electrostatic force, Po t<> P- The proportion of ions lost, t John Digitized by Google A lUDlO-ACTIVE SUBSTA^^Cfi, ETC. 2()1 imultaneously in the gas. From the results of the to conclude that the ions thus produced resemble tgen rays, and carry the same charge. We will rning the recombination will not be much different nding the correction for recombination has been [t was there shown that for small conductivities bout 4 per cent, of the loss due to diffusion to the ;s made with Rontgen rays was about Yb-th of a ig 1 5 millims. A new apparatus was made with without altering KZ^/a^V, the value of Zj/V is ons which recombine is similarly reduced from tained in a sealed glass tube, which cut down s to produce densities of ionisation less than the jriments made with Rontgen rays. We may experiments the process of recombination does of '5 per cent. ual Repulsion. n (as in the case of ions produced by the action )r by a point discharge), the electrostatic field letimes suflBcient to exert a considerable force, imount that this effect contributes to the loss ng place, but it is easy to find an upper limit I gas in a metal tube losing its electrification iiies of force from the axis to the surface. If )lace, it is easy to showt that the density of :he formula — Po 1 -f 4c'Trupf/ ' Liforiii, u the velocity of an ion when acted on time during which the density falls from practically AnpoUt when the loss is small. >, ' Phil. Mag.,* Juiie, 1898. Digitized by Google Dounda r down d by tl a^ to tl th air at he mova be of tl 3 tightly leading t( atus. W 3 tubes w( •elocity of gasometei 3 suljstanc r), and th< of thin gl f wire 8up] it by the ac through th( C was seale i radio-activ< ebonite supj haken when was eonnecte eing to earth. the other pa rth. The rod metal screens le electrometer abe A is in n ipt E, there is ii es T into the s]); dctrode, and a d ordinary conditi* turbulent moti< itial difference ( periment never < ionisation and \ larging A to 40 one sign are en :o the number of Digitized by Google F A RADIO-ACTIVE SUBSTANCE, ETC. 263 the disc. When either set of tubes was pushed arge tube divides itself equally among the twenty- bes T. The disc a^ was soldered to the front of the nt of the tube B^j. ospheric pressure the stream of air was obtained by ylinder of a gasometer. For experiments with dry asometer was connected to wide tubes of calcium ked with glass-wool was put between the drying so that particles of dust should not be carried into it was desired to make experiments with moist air, emoved, and long tubes half filled with water were air along the tubes T could be varied by changing as obtained from E. de Haen (Chemische Fabrik, eparation labelled " Radio-active Substance A " was containing some of the radio-active substance, was s inside the tube A as shown in the figure. The substance was much more intense than uranium rays, iss tube was strong enough to ionize the surrounding 1 order to prevent any moisture from coming into ibstance, which was deliquescent. The tube A was 5, S, to the top of a heavy box, so that the tube C tubes B| and IB^ ^^® fixed in position, o one terminal of a battery of forty lead cells, the i'he rod F was connected to one pair of quadrants of f quadrants and the case of the electrometer being and the wire connecting it to the electrometer were \ that external electric charges should not give any tie. illic connection with all the parts of the diflRision electric force acting on a stream of gas until it comes between E and B. The air takes about one second 3rence of potential of a few volts between E and B J, suffice to collect all the ions of one sign on E, but of the gas as it escapes from the tubes T, a much ) volts) was used. The potential of the electrode needed 1 or 2 volts. It was found under similar ocity of air that the electrometer deflection was not ^Its instead of 80. We therefore conclude that all ^cted on E, so that the electrometer deflection is )ns that come through the tubes T, Digitized by Google BY THE ACTiC nds of the tubes 1 tube A, and was A CQuld be found, apillary tubes K, le admission of ai ary tubing was coi made with wide t vessels W^ and W. er to obtain a streai between suitable lin] iusted until the pn ^as connected to tht IS adjusted so as to a open to the air. 1 as turned on for a 3re desirable necessil out by the water-pu f rose at the rate of a the pressure was as jnt The velocity V >f air that escaped fr 5X0V,— A. Digitized by Google N OP A feAbiO-ACtlVE SlJiBSTANCE, ETC. 265 I and Bjj. A short brass tube L was soldered near the onnected to the manometer M, so that the pressure of The air from the room was admitted to the apparatus laving first passed through a tube of glaiss-wool G, to Y dust which might alter the resistance of the tubing, lected to the drying tubes, and the rest of the connec- :bing. The tube u leading from B was connected to which were exhausted by means of a water-pump. ^£ ^i>^ 1 ~ g y Fig E. of air through A at a given pressure P, and with a s, the stopcock Si was closed and the whole apparatus mre was a few millims. below the pressure P. The elivery tube of a gasometer, the movable cylinder of 3 on the point of moving downwards when the gaso- j vessel Wg was then connected with the water-pump, V minutes. The velocities V (through the tubing T) ed a larger supply through the apparatus than could ) so that the pressure, as shown by the manometer, ut 3 milliius. per minute. The stopcock Sj was turned ich above P as it was below P at the beginning of the the tubes T can be accurately found by observing the the gasometer and the time during which the stop- 2 M Digitized by VnOOQ iC " THE ACT roa of men a the tempe jitive Ions ii V- 1 J 344 Vt 387 55 420 4C 410 30 682 20 itive Ions in V. P. 1 368 77: 430 40( 609 20( n bet^ icaUy I ^een t »y me? 1 ^ X Digitized by VjOOQ IC N OF A RADIO-ACTIVE SUBSTANCE, ETC. 267 ry ; V is the mean velocity in the tubing Tj in centime, ture of the air during the experiment : — Dry Air. Table II. — Negative Ions in Dry Air. a »i. na. V. P. e. 19 63-4 138-6 344 772 19 13 43-0 93-8 387 650 13 16 24-8 680 420 400 16 13 10-6 39-9 410 300 13 1 12 7-6 31-5 582 200 12 [cist Air. Table IV. — Negative Ions in Moist Air. a 18 11 9-5 «1. «2. V. P. e. 711 135 368 772 18 21-0 66-3 430 440 11 7-6 27-1 609 200 9-6 I ratio y {= ^'i/wg) and the coefficient of diffusion can 3 of a curve representing equation 2, Section I. The 9 i'O hi i'B i'3 h^ h^ 3 M 3 Digitized by VnOOQ iC BY THE ACT] •es, Si and S^ were A quartz-plate, e the joint air-tij Jaced in the wii ^ Digitized by VnOOQ iC ON OF A RADIO-ACITVE SUBSTANCE, ETC. 269 saddled on to it, each of them surrounding one of the J, was fixed to the end of S^ by means of sealing wax, ht, and a piece of wire gauze, having the same curvature dow Wi, completely filling it. A piece of zinc, Z, of the 3ce of brass which was cut out of the window Wg, was fixed sed through the ebonite disc D. The disc fitted tightly joint was made air-tight. The zinc did not touch the il relative to A could be varied as desired. When ultra- he quartz and the gauze, it falls on the zinc, and negative face of the metal. Some of these ions can be sent into a ^ A by lowering the potential of the zinc relative to A. battery, H, was insulated and its positive terminal was .tive terminal to R. minium wires was used as the source of ultra-violet light, g the spark was contained inside a box covered with lead, through which the light from the spark fell on the quartz- nals of the secondary of a Ruhmkorfi* coil was connected joyden jar, and the other terminal to the inner coating, e coil, and the discharge took place across the spark-gap ium wires. The air in the neighbourhood of the spark id, so that it was found necessary to pack wool round the rifled air from coming into the neighbourhood of the rod F : to the electrometer. When this precaution was taken it Digitized by VnOOQ iC BY THE ACT The density of ionizatioi has to be polished from ti smalL In the above exp the third experiment with gas in the tubes T^ is 5^7 apparatus per minute was On standardising the el sponded to a charge of Oi fication p was therefore { Section L that the product due to self-repulsion shouL the sides. In the present need be made for the loss c Secttion I\ In order to make the a charge, the changes shown diameter) were made in th< F%g.4 ^h 'h Digitized by Google OP A RADIO-ACTIVE SUBSTANCE, ETC. 271 pends greatly on the state of the zinc surface, which bo time in order that the ionization should not be too ents the greatest density of electrification occurs in ga-s. The mean time, ^, spent by any portion of the ad. The total volume of gas that passed through the > c.c. ometer, it was found that each scale division corre- electrostatic unit. The mean density of the electri- < 10"^ electrostatic unit per C.C. We have shown in C t must be less than 10"* in order that the loss of ions B less than 1 per cent, of the loss due to diffusion to e the product pX* is '9 X 10"^\ so that no correction to self-repulsion. -Ions produced by the Point Discharge. stratus suitable for experimenting with the point dis- L fig. 4 were made. Two circular holes (1*6 centims. in ibe A, and two tubes, Q and B, of the same diameter as ;,£^- Fig. 6. w Digitized by VjOOQ IC \r * - • -S.^:^-- as 1 J>anie as iitinis, fn»i xjx^riineiit burth ex J I :es place i Digitized by VnOOQ iC S OF A tVPKV.ACriYK srRSTANCR KTX\ l^7S I stcvl \\i\\t in the tiiK* Q, the jxMiit Knr*g at :V»e e ui:der5«ine ivndit unis as exj>oriment K exivpt tV,ai • :\iVr that the ::[5u^ v^houM have a siiialler eUvtnt;vxu:ov, vriinents 2 and 3, with a platimim jvnnt sul^titiuoil .< held in the tul^ A hi the [xv^tiou shown ux ti^j. o, ermient 1, except that the }x>int n>-5U5 dniwu up tho 111 the aperture iu A* »iily ones iu which the efteot itf st*lf-n*pxilslon may ;s of ions in the tube Tj, so that the n^Uuos of K lav be a little too big. *ent. which occurs lietweeu the vahuvs obtainiHt in »s obtained iu experiments 2, 3, and 4, is pix>lvibly ifference in the ions, iiu of air in A from a |x>iut some distance up tho 5, as Experiment 6 sho\>T5 that they dittuse nunv -Xegati ive Ions in Dry Air. «i. fi^. V. K. ■2 138 337 •0382 62 165 326 •0367 <-5 I 150 1 323 ' -0368 ■2 i 1 I 160 342 •0324 [)oiiit ill the tul)e Q, tho point Wnug at tho 1, with a platinum point Hul>Hlitutod for [\\v I the tube A, as shown in fi^. 5. 7, exct»pt that the j)oint was drawn uj) \\\\' ii-e ill A. cjilly the Hamo vahioH for tho ooollicioni of vH that larger ions aio imxhuvd when llio tube Q. Digitized by VnOOQ iC BY THE AC The theory of the inter iversely proportional to tl een confirmed by the exjv Section IL show that the 1 onsists of ions. The prt ontribute to the total pn ressnre in this case is the :e see that between the iffiision is inversely propo We conclude firom this t aries between these linlit^ L ' The experiments on diti'i ctive substances, and ultn he same changes arising oefficients of diffiision of i ases which are greater thu The ions produced by th ther methods, since theii egative ions in moist air. Coefficients of Dit- Method. Rontgen rays Radio-active siil>stance Ultra-violet light Point discharge Me^ Digitized by Google OF A RADIO-ACTIVE SUBSTANCE, ETC. 275 IX v.. of Pressure. an of gases shows that the coefficient of diffusion is 1 pressure of the two diffusing gases. This law has ts of LoscHMiDT and others.* The results given in 1 be extended to the case where one of the gases that the ions exert is so small that it does not \y an amount which could be measured. The total ire 0^ the gas into which the ions are diffusing, and as 772 and 200 (millims. of mercury) the rate of ;o the pressure, size of an ion does not change when the pressure iced hy Various Methods. w that the ions produced by Rontgen rays, radio- ight are nearly of the same size, and subject to ) presence of moisture. The following table of lir shows that there are differences in the various light arise from experimental errors, ischarge are larger than those produced by the diffusion is much slower, except in the case of ons produced in Air by different Methods. Dry air. Moist Air. ) ions. Negative ions. Positive ions. Negative ions. •035 •043 •032 •043 •036 •041 •043 — •037 1 •037 •032 •028 •027 •039 •087 Theory of Gases,' Chap. VIII. ? N 2 Digitized by VjOOQ IC [E AC y that It of tl > come iits of ^ woul< ficient? Digitized by VnOOQ iC OF A RADIO-ACTIVE SUBSTANCE, ETC. 277 sizes of the ions produced by point discharges vary iparattis in the neighbourhood of the point. What is definite conclusion with regard to the charges is to sion and the velocities of ions produced under similar possible with the apparatus I have used for the deter- [iflEusion, and I hope to be able to make observations to an accurate determination of Ne. charge on an ion in a gas in terms of the charge on is of some importance, as it enables us to obtain f electricity. IS a similarity between the minimum subdivisions of gases. r the determination of the charge in absolute units b gases, and since all the determinations depend upon ^reat accuracy cannot be expected. The results show is of the same order for ions obtained by various )een obtained by Professor J. J. Thomson for ions Lnd by ultra-violet light ;t the values are nearly the ^ and 7 X 10"^^ electrostatic unit. These values do due 5 X 10~^^ which I obtained for the charge on the en off by electrolysis.;}; ;e is the same in all cases, we must assume that the in order to account for the differences observed in the LELLAND,§ by examining the velocities of the ions owing wires, found that the mass attached to the ion circumstances connected with the ionization. The es for small differences of temperature of the wire, llects round an ion is very variable. We would not dio-active substances would have an effect upon the icy to collect round a charged ion, but it is possible Afferent ways by different kinds of rays, so that the h point discharges in air there are actions taking e the carrier increase in size. Thus the oxides of t condense round the charge and lower the rate of of the ion. -violet light has any effect on dry air, but Wilson|| Phil. Mag.,' Dec, 1898. Phil. Mag.,' Dec, 1899. * Phil. Mag.,' Feb., 1898. 1), * Camb. Phil. Soc. Proc,' vol. 10, Part VL * Phil. Trans.,' A, vol 192, 1899. Digitized by Google 279 ] %re of Metah, (Second Paper.) Mechanism and Applied Mechanics in the TER RosENHAiN, B.A., St. John's Colhge, h Scholar, University of Melbourne. ead, in Abstract, May 31, 1900. :s 3—13.] 3resent paper deal principally with the >ntinuation of the research described in the ' A, voL 193, 1900, pp. 353-377). In iron, •een studied with the aid of the microscope D, Chakpy, Stead, and Roberts-Austen esult of their labours it is well known that nent of the crystalline grains of the metal. 1 in tension its crystalline grains become ; when the specimen has been subsequently all signs of such elongation disappear from microscope. * In fact it is not generally Btween the crystalline pattern seen in the ling. In general, the pattern seen after ilar specimen before it has been strained, em- produced depend very much on the trly upon the temperature applied, the time g. Arnold and Stead have shown that rge crystals in iron and steel. But even e is well known to produce complete re- that these changes occur at critical points the cooling of the metal. These arrest- is natural to suppose that they are evi- of the metal. ! hoped to observe this change taking place jrimental difficulties of keeping a specimen J being heated were successfully overcome, of iron failed. 10.12.1900. Digitized by VnOOQ iC ^E STEUCTUEE OF METALS. 281 r the atmosphere. On leaving the apparatus ugli the window by means of " vertical " illu- ve itself; as we were content with moderate n objective of long focus could be used. , we did not succeed in keeping the polished » had been reached ; but in the course of our )n was observed. On beginning an observation, '' ferrite " grains could be clearly distinguished. t'^ere then slowly raised by gradually turning on irst visible change was a dimming of the image, etely blotted out. This we supposed to be due 3 part of the optical system, but we could not re further, the image of the crystals reappeared ting off the reflected light, the metal could be mating still further, the pattern was rapidly and 5 of the surface ; the metal was now dull-red. ily dark spots appeared, and spread rapidly over eed at which they spread could, however, be leating current. The spots appeared well in the )parent darkening could only be pushed to the erably higher temperature. On allowing the isible, either on passing through this range of i ; nor could the phenomenon be made to recur d below redness; but, if this was done, the y in the same specimen. It seems probable that ance occurs in the metal itself and not merely in Lis in this film remain entirely unaffected by it, 'ssion that he is looking at an action taking place arent film. On repeating these observations with atmospheres than hydrogen, no such phenomenon to suppose that the phenomenon is a result of '^drogen and the iron. From its occurrence just sspond to the arrest-point, about 487^ C, discovered ^s Research Report,' Inst. Mechan. Engin., 1899). hydrogen caused the surface of our specimens to rercome this difficulty by observing the surface of a ating was again done electrically, either by passing ecimen, or else by placing the specimen in the centre and on a piece of terra-cotta. In both cases the , the electrodes passing through a sealed cork at the 2 o Digitized by Google ^E STRUCTURE OF METALS. 283 cloned the attempt to observe this process iu to the study of similar processes of amiealing iible metals, particularly lead. -ture required to produced re-crystallisation in we observed in specimens of plumbers' sheet- ite nitric acid. When thus treated, ordinary Laiit crystalline structure on such a large scale The etched surface shows all the appearances light on etched crystalline surfaces ; when the he various crystalline grains in turn, the colour each grain, but different on different grains. ich a surface magnified two diameters : these ring, and must therefore be observed and pho- ►lution. •face reveals a peculiarity in the configuration of jn to have many remarkably straight boundaries of parallel boundaries being frequently observed. ible what we had previously observed in wrought [uent occurrence of twin crystals. In our earlier presence had always been readily detected by the in them by slightly straining the specimen after observed in sheet lead by this method has been Phil. Trans.,' A, vol. 193, 1900, Plate 26, fig. 40). of detecting twins is not available, as the rough- iepth of etching employed make it impossible to mce of twin lamellsa nevertheless becomes evident with oblique light. Fig. 3, Plate 3, is a photo- B magnified 40 diameters. The figure illustrates imination, which has picked out a few isolated J while neighbouring ones remain almost dark. luminated grains, a number of dark patches are - boundaries occurring in parallel sets which are is instance there are three distinct parallel sets hey are twin lamellae becomes apparent when the .ted, thus altering the incidence of the light. As Qs that were bright become dark, but presently eviously dark shine out brilliantly, all the bands ashing out simultaneously. Fig. 4 is a photograph rotation of about 30°, and illustrateis this appear- '^hich catch the light simultaneously are evidently which the orientation of the elements has been other words, they are twin lamellsB. 2 o 2 Digitized by Google CRYSTALLINE STRUCTURE OF METALS. 285 ermine the effect of very severe strain on the crystalline a soft, ductile metal, plastic deformation may be carried to le adaptability of the individual crystals to change their leavage planes may be insuflficient. Careful observation of f a piece 'of lead under severe compression confirms this it the crystals are gradually flattened out in proportion to specimen, but when the " flow " becomes considerable it is ady very thin and flat, are driven into and through one ig in a grain or structure which is small, but still entirely inalogous to what occurs in the fracture of a more brittle ;hat in a more brittle metal, when " slip " has gone so far rystal, the new siu'faces thereby brought into contact do cture results ; in lead the freshly exposed surfaces do weld re, a fact which is associated with the possession, on the utility. Fig. 6 is a micro-photograph showing the crystal- i lead magnified 1 2 diameters, while fig. 7 shows the much 3shly and severely-strained lead magnified 30 diameters, xperiments with lead, the process of straining was carried of the metal in a compression-testing machine, letting jlock, originally about 1 inch high and f inch diameter ut I inch thick. e changes in the crystalline structure of such strained of taking a series of photographs of a marked area of ;ime during which the metal was exposed either to the room or vras subjected to special thermal treatment. iken the surface was thoroughly re-etched ; our experi- L had convinced us of the necessity of this proceeding, pecimens have confirmed the previous experience. In t in any way produce a visible change in the surface ad been resorted to, and fairly deep etching is required tirely. This applies more particularly to the channels ater-crystalline boundaries — ^these may often be seen y-formed pattern, but quite independent of the new ted of alternate applications of concentrated and very , where very deep etching was required, an electrolytic great advantages of dealing with a metal like lead jrystaA& ; by enabling us to use deep etching it allows sed vsrith, and it becomes possible to obtain micro- ns, and under oblique light, which exhibit clearly the Digitized by VnOOQ iC LINE STRUCTURE OF METALS. 287 erefore be taken as no more than an extremely agnitude with which we are concerned in these iperatures. sheet-lead showing fairly large crystals as an perature annealing has continued for a long time, plying a higher temperature, so as to determine occur. Our observations show that the metal in •ately high temperatures, three minutes' exposure ture of 200° C. being suflficient to produce a great If the specimen be kept at 200° C. for a long time it becomes very slow, and ultimately a state is »erceptible. inclusive are a series of micro-photographs of a 1 4 shows the appearance of a typical specimen of iange produced by 30 minutes at 200° C. Except is very diflficult to trace any connection between ginal. Fig. 16 shows the same surface, re-etched )^ C There has been further change, but not to n the first half-hour. The change is most marked side of the figure ; in fig. 1 5 it shows a mottled or nes filled in in fig. 16, while there is a considerable ss of the two tongue-like projections that start on nd. crystal is seen in fig. 17, which was taken after ^ C. Here another twin band has become evident, htening of the boundaries has taken place. This n fig. 18, taken after four days' further annealing. representing the final state of this specimen, as 1 no further considerable change. This specimen, ng feature, which we have often observed in other )r photography occupied the centre of the surface 1 approximately f inch square by ^ inch thick, marked area did not show by any means the best 1. In this case, as in many others, we found that ing crystals were formed at or near the edges of it the same stage as fig. 18, and with the same irked area, shows the remarkable development of men. uch a series of photographs, one consideration must 3nce produced in the appearance of the surface by f incidence of the light. In spite of the utmost Digitized by VnOOQ iC CRYSTALLM STRUCTURE OF METALS. 289 men; the following photographs were taken after the to 200^ C. for the time shown in the table : — amber. Days. Hours. 21. . . — 17-5 12. . . I 17-5 3. . . 2 17-5 4 . . . 5 16 5 . . . 39 20 1 . . . 39 20 icture characteristic of freshly-strained lead, with one ave persisted from the original crystallisation. In rrown considerably, and a general change of pattern g feature is the large skeleton crystal that hafi / corner of the marked area. This skeleton is seen 1 figs. 22 and 23. Figs. 24 and 25 were taken under in order to show another large crystal which gave ig. 24 it is still somewhat skeletal, but in fig. 25, 3onsolidated, all its outlying arms have disappeared, lefined crystal, part of which is seen as a dark arm h of the specimen at the same stage as fig. 25, but jrs), and so illuminated as to bring the new crystal, , into brightness. This new crystal is seen to be irs, and from its position relatively to the marked ime crystal whose early stages are seen in figs. 22 3lJent example of what may be called an aggressive 3, also at 8 diameters, is shown in fig. 27, Plate 10. men can be seen, and the photographs illustrate stals are generally near the edges of the speci- 3 Jarg-e crystals are not mere surface layers, but ess of the specimen, and can be readily identified s case, the specimen is a plate about one-eighth J in the annealed metal is apparently in no way 3,1s in the original state before straining ; the in a specimen whose original crystals were by the photographs (figs. 26 and 27) of these ce of twin crystals, both as inclusions in the 3S. In fig. 26 three distinct sets of straight 2 i> Digitized by Google rSTALLINE STKUCTURE OF METALS. 291 th oi ciystals occur in lead only when the metal has )vere plastic strain. The structure of a cast specimen ures which cause a strained specimen to show rapid i arranged to cause rapid cooling, specimens of lead lute crystalline structiu'e, whose scale is not very Tushed lead ; such a specimen was exposed to 200° C. sibJe change of structure occurred. A piece of this severe crushing, and on further exposure to 200° C specimen is cooled from temperatures of 200° C. to r-temperature has no visible effect on the structure. has no visible effect ; quenching in water, cooling in were all tried on a number of specimens without 3ned to a small extent by severe strain, and the ^storing softness is correspondingly small. In one ^ad was crushed under a given load in the testing- mtil no further creeping occurred. The specimen i under the same load, when a distinct amount of lace. )ed above as having been made with lead were nt lend themselves to similar treatment ; those well shown when a surface of a cast ingot of the iJoric acid. These crystals are generally large, s obtained on etching the surface of commercial nter-crystalline boundaries may be seen on the e grooves or channels. The presence of these 16 method of manufacture, during which these ted tin, and allowed to drain. Ab the plate is it crystallises, but any fusible impurity present ' longer, and, being forced by the crystallising s, the still fluid impurities will drain off, thus i of comnciercial tin-plate is shown in fig. 28, lalf the natural size. In this photograph the irly seen, but it also illustrates another and ixll ca^es of an etched crystalline metal viewed v^ed that, under a given incidence of light, ^were dark, and that the illumination was ;tal. In the etched tin-plate this is not the l» 2 Digitized by Google TALLINE STRUCTURE OF METALS. 293 f of true crystalline plates ; such distortion would in the coefficients of expansion of iron and tin rate of cooling. Considering the extreme thinness t of distortion might well be purely elastic and crystals of tin. specimen, but to the same scale as fig. 31, illustrates and arrangements of the tin crystals that can be ^ solidification ; the crystals in fig. 31 were formed >se in fig. 32 by quenching the specimen in water ^as still melted. By means of local quenching and attems can be obtained ; such processes have long nufacture of what is called *' moirSe metallique." it the small crystals of tin which are obtained by water do not show any growth when the metal is ?mperatures short of the melting-point. Even a does not make them grow or re-arrange themselves, a', be reduced to a minutely crystalline structure by lens so treated we have observed re-crystallisation to ts on the re-crystallisation of cadmium at moderate can be strainec^ by compressiou until its crystalline gh interpenetration of the original larger crystals. (12 diameters) photograph of an etched and marked strained piece of cadmium. Fig. 34 shows the same exposure to 200® C. It now shows a well-defined hows the same area again, after six days' further considerable increase in the size of the crystals is ;he gradual growth of some of the crystals is very itures that we have observed in the case of lead are 1 we can see no invading branches and no aggressive ;o be any considerable amount of twinning, just described were also made on specimens of zinc, * zinc strained by compression at ordinary tempera- on exposure to 200*^ C. Some results obtained with electric batteries, were particularly interesting. It lechanical properties of zinc are widely different at ly that the metal is soft and ductile at temperatures it is generally worked at that temperature, while it at and above 200'' C. Commercial sheet-zinc, rolled emains fairly soft and flexible at ordinary tempera- e is too minute to be seen in specimens etched with- Digitized by Google TALLINE STRUCTURE OF METALS. 295 has been very widely believed that annealing or iron and steel, are "critical" phenomena which can ^finite temperatures. Arnold has gone so far as to such an " annealing point." Various of the " arrest- and steel have also been regarded as representing Baling, but the connection between the two is by no uud phenomena of annealing or re-crystallisation in e interesting ^o inquire whether any corresponding the cooling of these metals. We investigated the trie arrangement consisting of two thermo-electric ionval galvanometer, and a potentiometer somewhat OBERTS- Austen ; the deflections of the galvanometer us of a telescope and scale, instead of being photo- )e, therefore, that either from this cause, or from whole arrangement, some minute arrest-points were Iting-points and the ordinary temperature of the air observed in the three metals tried, i.e., lead, tin and f they exist at all, may be found at much lower 1 our experiments were carried, te the phenomena of re-crystallisation in lead, &c., 1 heat is evolved during the cooling of the metal, xt even in iron the arrest-points are not necessarily ing, we look for a theoretical explanation of these he theory of re-crystallisation which we shall now s for the explanation of the phenomena described lit part in the action to the impurities present even ies which we believe to be of importance are those utectic alloys, or fusible compounds, with the metal nainly metals, particularly the more fusible metals, , mercury, sodium, or even rarer metals, such as when a metal containing a small proportion of such [ties are, for the most part, segregated in the inter- •^stals themselves form at a temperature when the lid, and the growing crystals gradually push the laries. Where the quantity of impurities present is n be seen under the microscope forming an inter- where the " pearlite " plays the part of a eutectic, ^ucture ; other examples can be found in the gold- ^ Messrs. Heyoock and Neville.* Where the very small, the meshes of inter-crystalline cement YCOCK and Neville, *Phil. Trans.,' A, vol. 194, plates 4, 5. Digitized by i Google ^ALIJNE STPvUCTURE OF METALS. 297 \9 over different crystals, in such a way as to produce een adjacent crystals. Another phenomenon, seen ic acid, is also of interest in this connection ; it has esent paper (see p. 284). We there have a case of and deposited upon another crystal in its proper ik it must be admitted that different crystal faces, heir elements, differ in solubility in the same solvent. difference is a further step in speculation which is, connection. Such differential actions may, however, differences of electrical potential in the surfaces w of the matter, then the diffusion across films of olysis. Now, while diffusion in metals and alloys is ectrolysis in an alloy has not yet been demonstrated land, the close analogy with salt solutions leads one ctrolysed, and those who have experimented in the rtain that greater experimental resources will not 5 phenomena of re-crystallisation which the solution oes not cover, while the electrolytic theory explains > fact that only strained crystals will grow, while ency to change even at higher temperatures. The eory, is that in the unstrained state the crystals are uous films of eutectic, and that electrolysis only itortion has broken through these films in places, ne into contact ; the electrolytic circuit would then :>ne crystal to the other by direct contact and back *e-crystallisation in solid metals may be summed up I is one of solution and diffusion of the pure metal > fusible and mobile eutectic forming the inter- results in the growth of one crystal at the expense n solubility of the crystal faces on opposite sides of bable that this phenomenon of directed diffusion is :iKR (*Comptes Eendus de TAcad^mie des Sciences,' vol. 116, Dii into iron is affected by the action of an electric current. He ► iron electrodes enclosed in a fire-clay tube ; the whole was iperes was passed for three hours, when the anode was found idorgone considerable cementation. This action in the interior carbon-iron eutectic. 2 Q Digitized by Google ALLINE STRUCTURE OF METALS. 299 of the weld and there ended quite abruptly. It that this weld line was mechanically weak; it icult to cut or tear the metal along the weld as in erefore behaved as a true inter-crystalline boundary, tectic, and therefore forming a barrier to crystalline earance of such a weld in section after annealing 30 diameters. The line AB is the weld. As these Qiercial lead, we were prepared to find that, as a eutectic would have occasionally found its way into seems to have happened only very rarely. We nd only in two instances did we see a slight amount 1 line of the weld. We think that we are justified IS to the accidental presence of impurity. If we have in a welding surface an inter-crystalline crystalline growth owing to the absence of eutectic, )lied, growth should occur there as elsewhere, erpose a thin but continuous layer of lead-bismuth in welding ; the specimen was then annealed for above the melting-point of the eutectic — but on the layer of eutectic had persisted as such, and lit in this case the film of eutectic introduced at the litions were therefore analogous to those which hold ystals, where, as we have pointed out, growth does nent conclusive it was necessary to have a discon- weld. We accordingly tried another experiment, 3S of the same alloy, and after aimealing we found ►ssed the line of the weld in many places. This times, various impurities being used, such as the c, pure tin, cadmium, bismuth, and mercury. All )wing considerable growth across the weld after t the amount of growth observed varied very much. 3 of crystals that have grown across the weld ; the iicated by a discontinuous line, CD, probably repre- i non-metallic character, around which the crystals ound the slag in wrought-iron. I to the action of the impurities which were intro- ite the possible contention that their action was of the nature of that of the " dirt " more or less ns, certain further experiments on welds in lead introduced at the weld was — 2 Q 2 digitized by Google TALLINE STRUCTURE OF METALS. 301 of a single specimen of freshly crushed cadmium, continued exposure to 200° C, under oblique light, • 12 diameters. (Plate 12.) sheet-zinc by exposure to 200^^ C, oblique light, ers. (Plate 13.) I weld in lead, using clean surfaces, after prolonged r the weld is seen at AB. Oblique light, magnifica- late 13.) I weld, using eutectic in the weld, after prolonged >f the weld is at CD. Oblique light, magnification 13.) Digitized by VjOOQ IC Phil. Trans., A.^ Vol. igs, Plate 4. -Etched sheet-lead, x 100. Fig. 7. — Freshly crushed lead, x 30. Digitized by VnOOQ iC Phil. Trans. ^ A.^ Vol. ig^^ Plate 5. Fig. 9. — Same after six days. \ Fig. 11. — Same after two months. Digitized by VjOOQ IC Phil. Trans.^ A.^ Vol igs^ Plate 6r Fig 13. — Same after six months. \ Fig. 15.— Same sheet lead. X 12. 30 minutes at 200** C. After Digitized by Google Digitized by Google ■> c. Digitized by VjOOQ IC Phil. Trans. ^ A.^ Vol. igs^ Plate 9. Fig. 25.— Same after 40 days at 200^ C. is; Fig. 25 (after 40 days at 200^ Digitized by VnOOQ iC 1 ^s n Phil Tram., A.^ Vol igs^ Pl<^te 10. nnealiog. X 8. Pig. 29.— Tin-plate, etched. X 100. Digitized by VnOOQ iC Phil Trans., J., Vol igs, Plate 11, I. X 100. Fig. 32.— Tin-plate, after re-melting the tin and cooling quickly, x i. Digitized by VnOOQ iC Phil Trans.. A., Vol. igs, Plate 12- X 12. 35. — S>ime after seven dajs at 200° C. Digitized by VjOOQ IC Phil Trans,, A,, Vol igs, Plate 13. 0" a X 8. D g-. 38. — Weld in Lead, with eutectic. x 30. Digitized by VnOOQ iC 03 ] I in a Magnetic Field. D,y F.R.S., and Alfred Hay, B.Sc. Read June 21, 1900. 14-21.] jrimental and partly mathematicaF, has isional cases of magnetic lines of force low of a viscous liquid.* The original ms made, showed that the stream lines estion, gave results very similar to those r the cases of an elliptical and circular at the stream lines imder these circum- direction of the corresponding magnetic ould be used for many practical investi- long research dealing with the various I extremely laborious, extending without a ise some method by which a thin sheet n could be obtained of any required veen two sheets of glass, the required e formed. w8 connecting the thickness of the thin rough it in a given time, so that the g to the differences of permeability of pertained. s undertaken of some cases suitably evere a test as possible for ascertaining i for any case, accurately, the position Ic field. ling in succession a very large number .n ellipse with the major axis parallel Lssociation Report ' (Section A), Bristol Meeting, 20.12.1900 Digitized by VnOOQ iC ^ LN A MAGNETIC FIELD. 305 iracteiises the method, is the fact that it the lilies of induction not only in air, but tei'ial itself. This cannot be accomplished 1 has therefore been applied to determine for a number of cases of mathematical plotted by calculation. Inasmuch as the f two liquid films with the corresponding bove investigation been determined, it is e which may be of interest, and some )f the method in cases of interest to icli has been necessary, has not — as far as tble in any published form, and although, it, parts of the problem have been dealt le account given will be found of use in of the relation between the thickness of e of flow. (Oil with theory. )btaining the stream-line diagrams. out mathematically and plotted, and its >tograph obtained by experiment, athematical interest, and also some of ;trical engineer. jstigation of the subject, and I elliptic cylinders and con-focal elliptic er A, tig. 1, from which it is forced by etermined by the pressure gauge B. It nt being secured by the lever D which E, ^Nvhere its temperature is recorded he slide J. ss fixed within the frame P ; a well is is introduced, a channel being formed Eii\ The correct thickness is obtained less gauges N, N, N, N, placed at the Digitized by Google IX A MAGNETIC FIETJ). 307 Film between Parallel Plates. 20 lbs. pressure. wing through in 10 minutes. to flow, minute. B. c. > 19 miuutes. 527 24 . 61-6 162 51-8 110 91 69-5 168 59-5 94-2 193-3 51-8 97-5 268-75 37 3 125-5 429 23-4 166-5 557 18 210 861 11-6 271 D78-6 9-28 320 172 8-04 329 !49 6-05 416 Digi ized by Google IN A 1 ^tAGNETtC FIELD. f ; / J t i / • J I J / \ / 1 ^ / *> / / ^. -■ ^ ■— B^ ILm in /V7C> n quduitii ftlNE 7 has. e. •O/ 5 I ■ A 1 1 / -S / 1 /' 1 / } / 1 i / J J r 7 ^ --- ■->. B_ in . I/Ve fnch n b 4dLn imt 'i ><0 809 Digitized by VjOOQ IC S^ A MAGNETIC FIELD. 311 en X — ^ty the velocity at the boundary )er unit width ot the layer of thickness 2 vdx Jo lin layer between parallel plane walls is ^er, seen to be proportional to the cube experimentally shows a satisfactory able us to calculate approximately the not, however, well adapted for exact it difficulty in measuring accurately the f the glass plates are either sufficiently eir great thickness) for refined measure- ibe of the thickness of the liquid layer, xtion of t win give rise to a large error 1 for the coefficient of viscosity in C.G.S. nding to a thickness of '012") the value and which may be accounted for either Y slight irregularities in the containing tre — all of which causes combined might :' the viscosity equal to 2*5 ; the density in fair agreement with the results of out careful absolute measurements, but to apply the method to two-dimensional ! curves obtained (figs. 2 and 3) were these experiments whs ordinary tap water, which [cation and Lubricants.* Digitized by VnOOQ iC IN A MAGKETIC FIELD. 313 4. Iflmpi 3 ew. D. view. O } ^lates and Clips. Digitized by VnOOQ iC A MAGNETIC FIELD. 315 ►vas taken to a place where it could be ire given to the sensitive photographic Zinc templates of the shapes required the wax, the outline was cut with a med. In cases where the flow in the )ned, and could only be removed after -s will be seen in the photographs, figs, ^ed at all. I. jtream-line method to the solution of decided to work out mathematically ction placed in an originally uniform 3 field, to plot the lines of magnetic Moo, and to compare the diagram theoretical diagram is given in fig. 9 agram in fig. 10. In making the com- 10 was prepared, and fig. 9,"**' which s then superposed on it : the coinci- )rily established the soundness of the \ that slight local divergences along stead of the sharp refraction of the leoretical diagram, we have in fig. 10 perfectly straight lines crossing the :• or smaller extent, in all the stream- It is more marked in those cases the two liquid layers, i.e., the perme- , is greater. It is clear that the l)s the originally uniform distribution comparison between the theoretical ary to assign to the liquid layer an hat of a particular stream-line in the lid in fig. 10 is clearly show^n by the tlie shape of their boundaries is the The method by which the solution [ fully explained in the mathematical boundaries of the liquid layer are means that the diagram gives the tion of this paper. Digitized by Google IN A MAGNETIC FIELD. 317 e this latter stream-line diagram with the 3nt in the shape and general distribution triking. im for an elliptic cylinder of permeability iced in a uniform field with the major axis :ing an angle of 45° with the impressed for a cylinder of the same permeability, . 23 is a similar diagram for a very thin How cylinder of fig. 16, but here turned ed that fig. 24 confirms the theoretical lollow elliptic cylinder bounded by two field be uniform. I may be treated theoretically as well as ses is very limited, and the vast majority beyond the powers of analysis. It is in ployed by us becomes a powerful weapon s of rectangular section, figs. 25 and 26 * square section placed with one of their o it respectively. In fig. 25, where the »ng the direction of the impressed field, ir are concave outwards ; in fig. 26, on a,pidly as we proceed along the field, the outwards. This suggests that an inter- x)und for which the lines exhibit neither Lnd, as a matter of fact, we know of one r. Fig. 26 closely corresponds to the cently succeeded in working out analyti- tiduction.* Figs. 27 and 28 are intended bh of one of the sides of the rectangular We know from theoretical considerations magnetising factor, and thus increasing very clearly brought out by a comparison 3tic fields corresponding to a cylinder of inside a hollow square one, and a solid teresting in connection with the problem s are slightly disfigured by air-bubbles, ^f, once they are allowed to reach a part film varies, ry, 1900, p. 225. Digitized by Google [ A MAGNETIC FIELD. 319 ^iical Appendix. 'ptic Cylinders and Confocal Elliptic tical theory ot magnetic induction was was the first to work out in detail the nagnetic material placed in a field of id his investigations to the case of an it a later date, gave an approximate nth its axis along a uniform field. In 1 ellipsoid of revolution placed in any I in solving the same problem for a A. G. Greenhill considered the case 72,\\ diagrams of lines of induction for erial placed in a uniform field. On ;ure in almost every text-book on the nown. In Maxwell's great treatise iagrams. These include the following •netised transversely, and placed with )ther (vol. 2, fig. 14); (2) a circular 5e direction, and placed in a uniform ;he cylinder is coincident with that of al in a uniform field (vol. 2, fig. 15) ; uniform field whose direction is at 1 of the cylinder (vol. 2, fig. 16) ; in infinitely long straight cylindrical ol. 2, fig. 17). iling with the induced magnetisation [e obtains a solution by assuming the B/r) cos (f>y where r is the distance of ^r, and (f> the angle between r and tants A and B assuming different substance, and outside the cylinder ^7'^ is an integral of the equation function Az + B^ of the complex t Ibid,, p. 66. § * Journal de Physique ' (1881). ! Magnetism,' pp. 493-495. Digitized by Google IN A MAGNETIC FIELD. 321 )s of the two media, and itj, Wn for the into the coiTesponding media, drical shells, it is convenient to abandon s, and to have recourse to the circular =1 (1) > — b^, represents the family of ellipses V r' > J/ > — a^, represents the family of = sin 6, ~ At f, - — =sinhw, [b- + I/) ellipse, and = constant to a hyper- t(> oo ), and (which may vary from >osition of a point in the plane of the nates. I potential function, we make use of -^11 that the magnetisation of a solid et one of the axes of the ellipsoid be lensional case of the ellipsoid to the d we see that in this case also the might be supposed to be produced inary magnetic matter, of volume- nt, to be displaced relatively to each I of magnetisation, such that pBs is the cylinder. If then pY stand for cylinders of the imaginary magnetic 9V isplaced cylinders is p -^ Bs. But Digitized by VnOOQ iC N A MAGNETIC FIELD. 323 v^ard-drawn normal at any point of the L becomes av.- putting u = tanh-i - we get for the (m-1)'>H (3). -.H. f. COS — (fi ^ l)ah sinh u cos 0} d from these equations. le cylinder is obviously ij = constant, il lines, we have to determine the ce the function which is conjugate te to sinh u cos is cosh u sin 6, — T — sinh?4 l)ao , K being a constant which varies nipressed field is along the minor >ii to the lines of the external field - , Sinn ?/, le line to another. elliptic cylinder may be extended Digitized by VnOOQ iC IN A MAGNETIC FIELD. 325 n w = u^, Vi = V^ ; this gives t + fcB, . (4) A^ cosh t^i — Atf sinh Uj (5). Si( ^ Bit ' . — jxbA^ — /jtaBj . . .... (6), I — fiA^ sinh u^ — /aB cosh i^j . . . (7) A, Ai, B^, and A^, we get ^cosh^i) , — im - l)a6(/Lico8hi^ + sinhi^^) * ^^ sh 2c^ 4- sinh Ui) , i \ TT l)aJ(yt6C0sh?<i + sinhwi) ' ^^ a somewhat simpler form by putting t^y 6i stand for the semi-axes of the ering that u^ = tanh *" ^ (&i/«i), we get TT • H ^a7}(-^) — Oft) - «i6) Digitized by VjOOQ IC N A MAGNETIC FIELD, 327 space, the substance of the shell, and external space. the view of plotting curves, it is con- ! cosines, such as the one compileil by ?al Society of London. Corresponding I easy step to pass to Cartesian rect- it for plotting the r^urves. mentioned, be extended to any number il, however, wlien numerical data are >uce in the equations instead of first lis may be regarded as a limiting case ^coming equal. If in the expressions e case of a hollow elliptic shell we II proceed to the limit h = a, we find i\'ith those deduced by Du Bois for a netic Shielding."* But although an cial case is thus obtained, it is much lells, to follow the method developed S Seddon 2, p. 613. Digitized by Google PhiL Trans,, A,, Vol, 195, Plate 14. 10 im-line Diagram corresponding to iheoretical Diagram of Fig. 9. 12 inite Elliptic Cylinder in uniform field, ^atio of axes 3:1. Permeability 1000. Digitized by Google Trans,, A., Vol. ig^, Plate 15. 14 Cylinder in uniform field, ability =100. 16 >tic Cylinder in uniform ieability = 100. Digitized by VnOOQ iC il. Trans., A., Vol. \g^, Plate 16. 18 iric Shield enclosing Solid . Permeability =100. 20 ar Cylindric Shield. line Diagram, -ability = 100. Digitized by Google Phil Trans., A., VoL 195, Plate 17. 22 r. Ratio of axes 3:1, major it 45^ to field. Stream-line . Permeability =100. '24: iptic Cylinder. lined 45° to field. >ility=^100. Digitized by Google Trans., A,, Vol. 195, Plate 18. of Square Section. lity=100. Rectangular Section. lity=100. Digitized by VnOOQ iC Trans., A., VoL 195, Plate 19. 30 of Triangular Section, ability =100. 3:i blinder enclosing Solid Permeability = 100. Digitized by Google 'ns.,A., Vol, \g^y Plate 20. ) and Teeth of Toothed - Permeability =100. 6 Lture by Pole-piece. Digitized by VnOOQ iC PhiL Trans., A., Vol. 195, Plate 21. 31) of Magnetic Field near edge of ole -piece in Dynamo with Toothed -core Armature. Digitized by Google \egrals to Optical Problems. \ Isaac Netvton Student in the Ige. HOMSON, F,RS. , 1899. Page notions 331 >red 331 ..,.•.... 331 ......... 332 333 334 335 ar a length of time which )ing detected .... 335 »e estimated by summing to which it is resolved 335 theory of widening of vibrations, Thomson's 338 by Lord Eayleigh. . 339 3f the pulses .... 339 o light of composition i pulse, subject to the the sDoiallest observable waves 339 itical expression gives 342 342 cZ hy a law 343 24.12.1900. Digitized by VnOOQ iC J. PROBLEMS. 331 Page 360 >chromatic light will arouse reasing intensity .... 361 [Kr's integral 361 some progress with the mathe- mpose natural light, e phenomena forbid us to regard iin of simple waves, such as may ne, the equations of optics find iesirable to enquire how far we )f simple wave-trains by means procedure was first suggested by permissibility of this process, and independence of the simple be found an attempt at a strict ciples (i.) that we are cognisant luced by the light during an the detector in use (the eye, a ned with simple wave-lengths, integrated energy we observe, as been put forward with great md plane-polarised light. :ter than by a quotation £rom srplications qu'elle donne des vement simple, dans lequel la • una Equation de la forme cements of an elastic medium, or of sctors, which can be interpreted in Digitized by Google :jal problems. 333 efining it for a finite interval alone. les of the time, from — oo to + oo. in nt)du fiy) sin uvch\ a f elementary simple vibrations, of Ht) results of the undulatory theory all values from zero to infinity, nber of simple circular functions have meaning, as in the familiar nstruments, &c. The question we g in the limit, when their number 1 ; this was, in fact, offered by ach of the component vibrations all time. This is true whatever ng. But this disturbance may, ? interval of time. Take the case I. PoiNCAR^, will separate the ed separately. Hence a spectro- time before it is kindled, and for Is must therefore be fallacious. spectroscope possesses infinite telescope will be illuminated by le matter from another point of rruji'ing in Physical Problems. — In pure Is subject to certain limitations. These luities, or infinite sets of fluctuations, or d neither infinities nor discontinuities. commonly used to represent physical t character of the method ; a function cjucstion would be without infinite or 95 ; also see Schuster, *C. R./ 120, Digitized by Google AL PEOBLEMS. 335 K - ^sM^- 3 upon the distribution of energy ■ phases of corresponding elements order to prove that, in case of a e amount of interference depends any assumption respecting the onsequences. •nfined to the particular case of perceptible fluctuations or other I the long waves of Hertz is by jreat compared with the periods sed to perceive and register the r chemical effect, photographic or minescence which the radiation le features of a single wave can ht, with a view to discriminating lely with the integral effect over isider the molecule as a simple istified, we may prove that the ling but the partition of energy :er integral. The phases will of it id. or no ; but, if that condition +• i/(«) depends upon ^f{t)dt — 00 qual to Digitized by VjOOQ IC AL PROBLEMS. 337 'erent periods drop out (a familiar ite of absolution is dependent on ►rptiou. So again with physiolo- s produced by light we probably le of luminescence also. It may * dissociation. But, if the disso- excited vibrations become large, ihtle88 some molecules will split iile the precise timing of its own be all-important. But on the f dissociation will perhaps depend ifying the average structure, linear equation for the vibration 3t step towards a solution of a >KES, ''Linearity applies to the } ether — but it does not follow omplex system of molecules." t treatment to the spectroscopic en direction is compounded of e of the instrument, and the ^ tff — depends upon the same ip — depends upon i/r.. rznining the constancy of the t is constant, so also is the iiy ; hence, the spectroscopic Digitized by Google PROBLEMS. 339 urve to completely specify the ts will be justified below. .EiGii* in his paper on "The proposes to regard this as an (8) (22) % is therefore C^e'^'^^du. necessarily equal to (8), and of le range from — oo to + cx), the value of u would be indeter- could be assigned ; and if the iltant would again be indeter- ; there is room for an infinite ioncerned only with an average, to be. proportional to the total phase-relations. In the aggre- ^y is distributed is still, for all theory. But when we come to are certain questions which still is confined to a certain small ssor Thomson's theory, how ;ive the properties which Lord Again, we might suppose that egrees of crowding among the b a great number of them are )e so thinly scattered as to be, occupied by each. Experiment 1 correct ; we may enquire what rd Rayleigh's theorem to the us, be equivalent to a spectrum, Digitized by Google PROBLEMS. 341 base of the resultant will be c motion A. lall, the new phases will differ leat, the new phases will differ 11 have no apparent connection radiation. First, we can only )h we have taken to be T ; we pulses in T. Now Schuster's dvgj of a radiation, and its ex- i-Ier expression for the resultant ion of V, but partakes of the this point we make use of 1 particular wave-lengths, but ave-length. Bearing in mind >es, we have a spectrum whose Lirve will be less rapid as we )ave just seen, when the time ms are so crowded as to be the meancurve n(f}%u). But if e of angle included in the set se intermediate between the e pass continuously to quicker ) and so on ; the divergences t on a broad and theoretically miount of energy in the slow )f Sound,' ed. 1894, p. 40. Digitized by VnOOQ iC niOBLEMS. 343 L much lai'ger exposure, or as are not all similar is obvious. es of constant displacement^ inie time the proportions of tween x and x + dx are, say, ulses ; suppose that they give * the mixture is jight. )n by supposing it to consist is practically a pulse of con- thickness of a pulse is com- r the cathode stream. Lord »e regarded as simple waves )v the properties of a succes- mcy of statement. Professor new. He has held that the Now this is a valid objection rht of any composition what- sess definite phase-relation ; )f the present essay that the truna of definite composition, of being brief in comparison 1 be a great number of them iverages over small ranges of es of time- and wave-length cession of pulses and that 3he single pulse. 1 the scale of Avave-length. of X which is zero from CO y, 1898. •, 1898. Digitized by VnOOQ iC >ROBLEMS. 345 ignetic force great and nega- legative are to be so balanced pulse shall vanish. On this IS, Sir George Stokes* bases be zero amplitude for the lergy in the visible spectrum wfU be of order E^d . {d/X^y, stead of 10"^. Now diffrac- j order ; very much shorter and in any case would give posed forms will be sensibly 3h higher degree than those IS, nber of independent pulses an incandescent gas. We I, and to have no visible tion as composed of plane- Furthermore, the amount ected by absorption. The free vibration, omogeneous. One reason les in the line of sight will doubtless be the altered This will perhaps become r it at present, iie train of single waves gth, but has a definite ed below. The Doppler arrive at the remarkable 3 pressure is indefinitely ted by Lord Rayleigh, ion virith another. The 3ction. The vibration >re be suddenly and otal radiation received )7. Digitized by Google PROBLEMS. 347 ig off from a maximum at k (i) ), where 7i is the distance from ave-length. cs both Athwart and in Line imum brightness, and definite erent lengths of train. v^ith velocity v, a fraction C-^'* 'Jr"4 ./p1 ("•). Jume, 3ity Vy a fraction of waves of length between crht. Lction ^e ^ dr^ will give free Digitized by VnOOQ iC L PROBLEMS. 349 hf^'^mi- igh the breadths of elementary •tly, yet the application of his yte detailed information. The retardation u of the two half eir " visibility " estimated for isibility-curve thus constructed m line, and find out something intensity of light for position x i-difference. nd 34. Digitized by VjOOQ IC PROBLEMS. 351 ble. We axe enabled, however, ves in his paper on the kinetic ,-p« < 4- 1) e-^dP notation. e values in column 6 of Tait's ^ the function is very near to •- 1 1 1 1 i 1 1 1 — 1 I 1 1 1 i r ^. \^ ^^ s. ^.:^ ^;>v '"'••., =^ ^ £-0 £•3 >se /x so that the two curves ; graphically we see that we ■^z/^2 > II. (dotted curve) is he integral through using 11. s of the order of the errors t^pare our results. Digitized by VnOOQ iC )BLEMS. 353 esent paper necessitates a tted from Lord Eayleigh's above-mentioned necessity theory. The conclusion of its for a certain fraction of rom f to ^ for the different iby I the spectrum on a scale of 11 be Sq. Hence the " half- onnected by the relation to the visibility half-width should be diminished by a md collisions comparatively LS would disappear, the free :pected that the formula for rd Rayleigh, in which the g is valid, there is no such indefinitely reduced; the It is noteworthy that the oes Lord Rayleigh. This tation. lathematically homogeneous certain continuous spectrum, yrsis that, for zero pressure, r curve for the total spectral Digitized by Google )BLEMS. 355 iomponent curves are, the lUow ourselves to substitute these become narrow, ied for r, will not affect the th of train is taken to be ater velocities give shorter )ther ; for it gives too great dties, which velocities send of the error is to make the he limiting width for zero , and then to proceed to the me. rum Lines. 3 vibrations of an atom are umple gas will be to some loped by JAUMANN.t ibration (x.) IS been pointed out in the ) physical meaning for the lent emissions of this form * a gas cannot strictly be (x.) will never be allowed a stage by a new collision. )f a naolecule has generally shall not be making a ives. We will proceed to ^s so rapid as to allow this application of Fourier's Digitized by Google OBLEMS. 357 mogeneous Light oducing a large number of f a certain regularity in the las been abundantly refuted )t be produced without the iges is an index, not of the )f the spectroscope, out a spectroscope, by using nogeneity. The number of case, it is also a test of the y a Fourier integral of the ill fluctuate rapidly in terms uency p^ we will use the values of u small compared ut + \lf)du. to a region on either side of - \lj)du with those of oospt. The Biiote a simple vibration of of the expression will be, [)ply to e Theory/' « Encycl. Brit.'; ' Phil. ^94 Digitized by VnOOQ iC MS. 359 ot is of the same order : =i. In accordance with 3^ wjwe-lenjjths ; while Hjueiicy of the light a naturtil nxdiatioii {see :er of this ti'eatmeiit of has a historiciU inteivst, may he worth while to vibrator is 8oon to dopond The effect of tlie irrogii- ^ning of the Rpectruni line, svhich laid the foundatioiiH TU) natural radiation ih a t of the irn^gularitit^H upon it of tlie natural vibratiouH ukritieH in the light would K 520. Digitized by VnOOQ iC [S. 361 iiiiig of the incident 11 of a large number of the amplitudes, nil of the component iiergy of the natural •ns ; in other words, will become greater there is no tendency interact one another. jes are, so to speak, IS light will give a e^being entirely due 1 be prevented from ponent trains would ation settles down to 5 irregularities, plication of Fourier. :egrand involves an re that the integral I- We are forced to s to the actual pro- he natural vibration the observed effects •n JR^rfu will excite Digitized by Google t/ Marble. rqfessor of Gex>logy in M.Inst.C.E., Head of lical Si'fiool {foi^ierly y). R.S. Page 363 369 370 370 373 376 jr . . . . , 382 tificial Deforma- 386 tificial Deforma- lighly contorted 387 398 I in many parts of the manner is a fact which a glance at any of the ich have been prepared the hardest rocks have ' or '* flow " of material e facts are undisputed, lug, has taken place is a 5.1.1001 Digitized by Google V OF MARBLE. 367 calcite, however, wais which existed in the ind cracks, as well as a finely pulverulent portions of the mass, jsure. GuMBEL con- ;ticity on the part of [)f the alahaster, and In the field, that the re they had become hows distinctly under to a recementation of *. It was shown by e in the case of the d to powder; in the ^ed that deformation for the phenomenon jliaracters and optical I been submitted to )ut by PFAFF.t He lofen in a steel block, V n upon it. A very estone, and this was e amounting to 9970 -as not displaced, and specimen of the same ure of 21,800 atmo- le limestone did not left on the polished the conclusion that y different materials, 5uts reproduces very ; described ; but the red out to receive a • the hole a steel die, ire, ■ Zeit. des Vereines Digitized by Google OF MAEBLE. 369 Jans of explosives. Tmmeter of about od together again chamber in which pJetely perforated ' the influence of what plastic, the e when removed, e marble had lost says, it was seen into a solid mass. )und the central of rocks (and it iJ work in which reared up to the 3 and in certain taken in any of applied, of the w cases, of the 5cription of the strength of the :ie pressure can ing about the s crust, where ssion will not ce molecular niovement to assure affects k ^virkt, der Digitized by Google OF MAEBLE. 371 mit combined with 1 however it was ^ tubes of wrought nanoe by rolling a le strip to the bar is then bored out, uarter of an inch the tube instead he requirements ) of the marble, in length, were ff, of Gottingen. ajs accomplished he tube, and so tube when cold. pass completely )n allowing the aed, and it waa 3red indispens- ; applied, as it le experiments >r immediately an accurately 3 was applied, ising a double accompanying ist of square )osite to it by 5 marble with being kept in to cylindrical , them when in diameter, las its upper 3tion to the rrosion and [na are cen- sures to the pper end of lure is kept Digitized by VnOOQ iC W OF MARBLE 873 ly bulge. This bulge rounding the marble. igns of rupture, when rratures. at ordinary tempera- Perent cases, the con- ns taking place more ninutes to 64 days, ceased, and in this The final amount of ved signs of rupture a close. imn before the pres- ) completion of the y slowly i the time ble, consistent with ;, and the tube was ling machine along as found to be still 5, now completely . without mechani- in between them, itting the marble ,. tube adhering to 1 the former case vhile in the latter or two instances le latter a smart ^ that it could be e exterior surface ;h and conformed all the fine tool y from the tube r the dimensions with equal case Digitized by VnOOQ iC V OF MARBLE. 873 Y bulge. This bulge 'ounding the marble, gns of rupture, when ratures. it ordinary tempera- erent cases, the con- s taking place more linutes to 64 days, ceased, and in this The final amount of red signs of rupture 3 a close. mn before the pres- completion of the y slowly, the time Je, consistent with , and the tube was ling machine along 3ts found to be still I, now completely without mechani- in between them, Itting the niarble tube adhering to 1 the former case ^hile in the latter or two instances ^e latter a smart y that it could be e exterior surface th and conformed . aU the fine tool y from the tube f the dimensions , ^ith equal ease Digitized by Google y OF MARBLE. 375 extent by the cones as above mentioned, h the unaltered cone en examined under h had taken plia.ce. r its turbid appear- aic of the unaltered in Experiment P of ved and branching through the rock. )andsof very small place. The calcite so produced have fter the movement mbers of irregular have been carried The structure is Ispars and many ble showing this ty and magnified '067 inch, which it bulging. The nulated portions itinctly twinned, movements of be seen to have attening of the have been bent it cases, whichi polysynthetio ihe destruction I of the calcite essure to alter lently due to 111 be referred of somewhaj; -this structure Digitized by VnOOQ iC OF MARBLE, ^77 ice (E). The whole Bunsen flame. The learlv {\s possible at le extreme limits i>f temperature !y hiH jKwgeiblo inn ill inches Digitized by VnOOQ iC OF MARBLE. 381 ut unfortunately it which rendered it n contact with the around the central 'here had been no ered marble tested When sliced and caclastic structure, 3k a foliation which ed the very narrow scribed. The twin by strain shadows, nor very striking, into very irregular irms (fig. 6), quite The individual lounced movement * twin lamellae, is 3 the calcite indi- ke outline of the twin lamellfe. In I inward between 1 interest, as it is over one another Digitized by Google F MARBLE. 383 •med while at a ier a pressure of slowly and at as 2 mouths. The Ui-ushing load after deformation. longer than the ori- ginal rock o 25*51 per cent, in former experi- Digitized by VnOOQ iC W MARBLE. 385 leformation is not 1 and some trans- ous mosaic before li section, starting >ward the middle, ones. Under the amber of fine and ilong which there vse, and elsewhere aclastic structure. )n may be said to e, accompanied by seen in the case 0^ C. The calcite al (none are more rmed rock a very flattening of the long as they are but no twinning, aning, givbg rise will show strain c twinning at the a when magnified lamellae in several of between '0005 I rock, it appears the calcite grains, above described, even in this iron- the deformation. 1 of infinitesimal hus contributing on, however, are Digitized by Google OF MARBLE. 387 1 three planes or Q to accommodate ids. The action is sUps, the portion olid. The process igate effect is not Lination in metals he crystal on one ) need to suppose dip when it occurs because the metal , and is able to be idividual crystals, its shape and its occurring within h those presented at the agi'eement 7 applicable to the Inscribed, as it is of :al is squeezed flat ARBLE BY ArTI- LlMESTONES AND J Crust. h s crust has been lalf-century, com- of limestones and Hy with unaltered ing from pressure, mites from many sections of these Neozoic age, does served, and states seen, producing a [>s, one of them a er K. Bayer. Akad. Digitized by Google OF MARBLE, l\H\) a. la. ^anada. Janada. , Canada. )tful Origin. SCructureif. la. la. I. uada. la. /anada. Digitized by VnOOQ iC L f k T y a ;d ELS of be ge he st- tits }tic % lOSt 'OW her i of ;up axe ^hat ight how In from ding med and rmed age ouble ( the labby ?en to )arely jalcite Digitized by Google f s t s e a a i i i s t« s 3 r a 1 1 Digitized by VjOOQ IC